Coherent structures are important in turbulent production, and are regarded as "beasts in turbulent jungle"^[1]. Over the past half a century, numerous efforts have been devoted in capturing and examining them in both incompressible^[2-9] and compressible^[10-14] wall-bounded flows. These structures manifest in a wide variety of shapes and sizes. However, no consistent conclusion has been presented on the elemental structure. In a low-speed boundary layer, typical coherent structures include single structures such as Λ-vortex^[15], soliton-like coherent structure (SCS)^[16], horseshoe or hairpin vortex^[17], pocket^[18], typical eddy^[19], and streamwise vortex. Further, there are composite structures such as hairpin packets^[20], low-speed streaks (LSSs)^[21], chain of ring-like vortices^[22], turbulent spots^[23], and very large scale motions (VLSM)^[24]. With the development in numerical simulations and experimental techniques over the last decades, the previously found coherent structures were confirmed and extended, especially by direct numerical simulation (DNS)^[25-27] and tomographic particle image velocimetry (Tomo-PIV)^[28-34]. Some of these structures are hairpin-like vortices, while others are non-hairpin-like structures including non-vortical wave packets. However, hairpin-like structures (including their asymmetric and composite variations) appear to be the dominant mainstay or the basic element in most of the literature, although diversified opinions exist from their shapes to their origins. This prevailing hairpin-based concept is sometimes applied crudely, which causes confusion and misconception of wall-bounded turbulent flows. In this paper, we attempt to clarify the concept of hairpin-based structure from its original idea to its forming mechanism. The new results from Tomo-PIV are presented to illustrate the characteristic of a three-dimensional (3D) wave, and the evolution and symbiosis of the wave-vortex are reported.

The hairpin-based framework includes all hairpin-like coherent structures regardless of whether they are symmetric or asymmetric, such as horseshoes, hairpins, cane-shape structures, arch vortices, and also their composite variations, such as hairpin packets or group hairpins. It appears that the cornerstones of hairpin-based frameworks are built from following fundamental hypotheses and classical experiments.

Theodorsen (1952)^[35] proposed a hairpin vortex model for turbulence production and dissipation in wall-bounded flows. In his hypothesis, a two-dimensional (2D) vortex line in the boundary layer moves away from the wall because of an environmental disturbance. Because the velocity away from the wall is higher than that near the wall, the middle part of the vortex line is stretched into a Λ-shape structure that grows outward from the wall with heads inclined downstream at 45°, with spanwise dimensions proportional to the distance from the wall, as shown in Fig. 1(a). This model was first identified by Head and Bandyopadhyay^[17] at a reasonably high Reynolds number, although the Λ-vortex was visualized earlier by Hama and Nutant^[15]. Most observed vortices in the turbulent boundary layer are asymmetric, and are predominantly distorted as arch-shaped structures, one-legged hairpins, or cane-like structures, as shown in Fig. 1(b). They are often observed above the buffer layer, and are considered as the result of instability of LSS^[36-37].

Fig. 1 Models of horseshoe vortex and hairpin vortex

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The first systematic investigation of coherent structures in a turbulent boundary layer dates back to the pioneering work of Stanford's group^{[21, 38-40]}. The prime tool in these experiments was a hydrogen bubble generating wire, mounted normal to the plate. They found that the turbulent bursts, which occurred frequently in the LSS, mainly consisted of ejection events with low-momentum fluids rising from the wall and swept down events with high-momentum fluids accelerating downward from the outer layer. This ejection-sweep process and the streak formation and breakup were explained by a mechanism based on vortex stretching and lift-up. The stretched and lifted vortices appear as nascent hairpin vortices. The near-wall bursting process was observed to be consistent with the passage of a horseshoe vortex by Offen and Kline^[40]. The early work of Stanford's group confirmed that coherent motions were critical in turbulent production in the near-wall region, and laid the foundation for the research on coherent structures.

It is noteworthy that the hypothesis termed as the attached eddy in the monograph of Townsend (1976)^[41] has inspired many researchers in understanding the physical insight of wall-bounded turbulence, and had guided some classical models^{[20, 42-43]}. The essence of the attached eddy is that its size is proportional to its distance from the wall. Townsend suggested a model in the form of conical eddies, and reported that other possibilities exist. This hypothesis may be one of the most important theoretical supports for hairpin models. As stated in Ref. [42], the Λ-vortex is a perfect candidate for Townsend's attached-eddy hypothesis.

The approach of kernel studies was applied in a turbulent boundary layer to examine the vortex dynamics of isolated flow structures^{[1-2, 44-47]}. Therein, hairpin vortices were generated artificially by a hemisphere protuberance or fluid ejection on the wall, and strong similarity was found with flow structures detected in bounded turbulent flows. Based on extensive and manifold experimental works, Smith and his co-authors suggested a model to elucidate the characteristics of LSS and the bursting behavior, which are closely associated with the evolution and regenerative process of hairpin vortices, as shown in Fig. 2.

Fig. 2 The conceptual model of hairpin vortices during a bursting process[1]

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Adrian et al.^[20] proposed that different sizes of vortex packets coexist in the boundary layer, and they are typically layered on the top of each other, as shown in Fig. 3. Small vortex packages fall in the low-momentum region induced by a large vortex. Large eddy packets are also wrapped by larger vortex packets in the low-momentum region. A uniform momentum zone is induced by sets of vortices that are longer than those in the low-speed region induced by a single hairpin vortex. Further, different vortices coexist in a vortex packet, and their velocities differ. The packets of hairpin vortices also constitute large-scale motions and VLSM in turbulent flows.

Fig. 3 The conceptual scenario of hairpin packet attached to the wall[43] (color online)

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Models centered around a hairpin vortex are supported by a number of simulations and experiments. Owing to space limitation, they are not listed herein individually. However, the mechanisms of generation and regeneration of a hairpin vortex are still not in consensus. They are primarily classified into the following three scenarios.

It is suggested that the origin and the generation of hairpin vortices are caused by the distortion and destabilization of low-speed fluids diffusing from the wall^{[1, 36, 39, 48]}. Smith proposed a model in 1984 to illustrate that the bursting process of an LSS involved vortex roll-up in the unstable shear layer formed on the top and sides of the streak^[1]. The lift-up and oscillation of the LSS were explained by an impressed local adverse pressure gradient, thus resulting in a local deceleration and a 3D inflectional profile. Once the inflectional profile develops, the streak will oscillate owing to local instability.

Inner-outer interaction is another mechanism suggested to explain the streak formation and its instability^{[37, 49]}. The streak instability was also emphasized by Waleffe^[50] in the model of a self-sustaining process (SSP), which was driven by the instability of streaks instead of rolls.

As mentioned in Section 2, Stanford's investigation interpreted streak formation and breakup based on vortex stretching and compression inspired by Lighthill's theory^[51]. Vortices were considered to originate from the stretching-caused intermittency and three-dimensionality of turbulence^[21], which was also supported by Ref. [47]. Further, they indicated that the lateral deformation of the vortex lines constituted the initial hairpin vortex whose head was observed to form at the interface between an ejected fluid and the inrushing outer flow in the strong inflectional region. Regeneration of hairpin vortices was further proposed as the result of the combined effect from a local surface pressure gradient induced by a parent hairpin vortex and an inner-outer interaction by the viscous-inviscid mechanism^{[2, 20, 52]}.

As first depicted by Hama et al.^[53], to describe fully-developed turbulence as a system of horseshoe vortices seems to be straining a concept which is the most valuable in the initial stages. They found that, before the appearance of vortex, there was a special structure called kink. Therefore, they thought that kink transferred to vortex^[15]. This kink structure was later identified as the SCS by Lee^[16], and the LSSs as well as bursting process were deemed as the consequence of SCS's behavior^{[16, 5, 54-60]}, as shown in Fig. 4^[5]. The hydrogen bubble visualization result in Fig. 4(a) shows that the SCS is formed prior to the hairpin vortex. Figure 4(b) indicates the model of SCS and its relevant vortices. When the SCS advects downstream at a speed of 0.6U_∞ ~ 0.8U_∞, an instability along the border occurs which then leads to a high shear layer. With the fluid inside the SCS moving upward, its surrounding flow will displace downward to cause the initial roll-up (the closed secondary vortex (SV)) which will further develop into the hairpin vortex (see the first ring vortex (FRV) in Fig. 4(b)). Chernyshenko and Baig^[61] also claimed that the LSS might not be the result of hairpin vortices. Thus, there is a possibility that active 3D wave structures live inside and constitute the LSS.

Fig. 4 The hairpin vortex model developed from the SCS[5] (color online)

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An experiment is conducted to study the evolution of the coherent structures using Tomo-PIV in an incompressible boundary layer over a flat plate. The experiment was conducted in the water tunnel at Peking University. The flat plate used is the same as Ref. [60]. Four CMOS cameras (1 024 pixels× 1 024 pixels) were used in the time-resolved Tomo-PIV, and they were arranged in a cross configuration, as shown in Fig. 5. A laser generator of Beijing ZK Laser Co., Ltd. (DCQ-30Q) was synchronized with cameras by highspeed controller of LaVision. The sampling frequency was set to 500 Hz, and the particle concentration was 2.3 particles/mm³. The final residual calibration errors for all cameras were below 0.1 pixels, which were further reduced by using volume self-calibration^[62] (to become below 0.01 pixels). The volume reconstruction was realized by applying the sequential motion tracking enhancement (SMTE) algorithm^[63]. A multi-pass approach with four steps was implemented in the volume correlation. Finally, the first four modes of proper orthogonal decomposition (POD)^[64] were extracted to analyze the velocity field.

Fig. 5 Experimental setup of Tomo-PIV, where four CMOS cameras (1 024 pixels×1 024 pixels) were used to measure the velocity in an area of 78 mm×78 mm

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A 3D velocity field data set was obtained by applying Tomo-PIV in a turbulent boundary layer at the low Reynolds number (Re_θ=420). By using a Lagrangian tracking method, the trajectories of a specific fluid domain are extracted. Thus, streaklines as well as time lines will be reproduced similar to hydrogen bubble visualization, which makes the repeatedly and freely multi-view analysis of 3D unsteady flow to be possible. Figure 6 shows two LSSs at the turbulent boundary layer, marked A and B separately. The spanwise spacing of A and B is Δz⁺ ≈ 97, which is in accordance with the previous observation^[1].

Fig. 6 LSSs in the near-wall turbulent boundary layer reconstructed from Tomo-PIV data sets at t=1.5 s, where the location of x+=0 is 462 mm from the leading edge

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The evolutions of 3D time-lines at the streaky region B are shown in Fig. 7. The flow is from up to down, and the arrows indicate the flow activity at the corresponding time. In Fig. 7(a), the time lines represent as a hump-shape structure, and then it evolves into the triangular bulge with the head lifted up, as shown in Fig. 7(b), which is very similar to the previous DNS result^[27]. As the bulge lifts up, the nearby flow will intrude into the center due to mass conservation. This lateral inrushing event and upward ejection event will produce a high shear layer at the interface. The edges of the bulge (dashed lines in the figures) begin to oscillate due to the instability. The initial roll-up will be induced at the border of the triangular bulge, which may develop into a streamwise vortex or a hairpin vortex. Figures 7(c) and 7(d) reveal that the triangular bulge will further evolve downstream precipitating subsequent breakdowns, and the bulge becomes asymmetric and variable.

Fig. 7 Evolutions of structures in the region of LSSs, where the arrows indicate the directions of the flow activity, and L represents lift-up (color online)

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In order to illustrate the conjecture of 3D wave, the initial domain is set as a flat surface located at different wall-normal positions covering the low-speed region B. Figure 8 shows the material surfaces at t =1.38 s evolving from different initial flat surfaces at t =1.2 s. They all manifest as wave structures spanning the near-wall region of y⁺ < 100. Figure 8 also reveals that the most violent upraising event occurs in the buffer layer and also the bottom of log-law region. These 3D waves emerge from the wall and develop upward into the edge of boundary layer with solitary-like properties. The advecting speed of 3D wave can be easily calculated from the positions of wave peak at each time instant. The average streamwise advecting speed of 3D wave is found to be approximately 65% of the freestream velocity. The width W⁺ of the wave form is nearly 80, the height H⁺ of the wave form is nearly 80, and the streamwise span L⁺ is close to 200. The dimension of wave structure is very close to that of the hairpin structure measured by Smith (see Fig. 2). The early work of Smith^[1] had noticed that the inflected region usually consists of several waves, and the number is generally from 2 to 5. However, these wave structures are usually ignored by most of researchers who focus mainly on the vortices. According to the works^{[16, 5, 54-60]}, these 3D wave structures may be identified as the SCS, which was argued to develop from a pair of oblique waves.

Fig. 8 Material surfaces evolving from different initial height levels (color online)

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It is known that there is no universal definition about vortices, and most of identification schemes on the turbulent boundary layer are based on the instantaneous velocity field^[65-66]. We try to further examine the flow by applying the Lagrangian detecting method. An objective vortices criterion called the Lagrangian-averaged vorticity deviation (LAVD) was proposed recently^[67], which is implemented to detect vortex boundaries in the present study. The LAVD is defined as^[67]

(1)

where X₀ is the initial material domain, ω is the vorticity, and is the averaged local vorticity over the domain of X(t; X₀). The structures detected by the tubular level surfaces of the LAVD are the Lagrangian coherent structures, which are the objectives to the coordinate frame. For a more thorough description about this method, see Refs. [67] and [68].

Figure 9 shows two quasi-streamwise vortices V₁ and V₂ by the LAVD method at t = 1.56 s. They are located at two sides of a wave structure, which is a material surface evolving from the flat surface initiated at y⁺ =27, and t =1.2 s. It is illustrated that the vortices emerge at the most inflectional region of the material surface, and they are not symmetrically distributed. Since no vortical structure is detected by the traditional Eulerian method in this study, we may conclude that the Lagrangian method has advantages in detecting the early vortex in the high shear layer. This is because most Lagrangian methods inspect and trace a domain of fluids for a finite time, which is easier to extract key structures with few effects from instantaneous noise. It is hard to read the cause and effect of wave-vortices only from Fig. 9. Most observations in the turbulent boundary layer find the symbiosis of wave structures, vortices, LSSs, and also other composite structures, and there must be the mutual induction among them. However, the saturate vortices are the most obvious structures to attract attention.

Fig. 9 Symbiosis of wave structures and Lagrangian vortices V1 and V2 (color online)

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The results obtained by the Tomo-PIV in Section 4 confirm the existence of a 3D wave in the turbulent boundary layer. In addition to the observation that a hairpin vortex dominates the boundary layer, the present study finds that LSSs with wave-like characteristics appear to be the mainstay in the early wall-bounded turbulence. Indeed, the hairpin vortex is a naturally produced structure behind a hemisphere protuberance on the wall, as depicted in Ref. [45]. The more important issue is what causes the natural turbulent boundary layer to behave like a protuberance-perturbed flow and why the wave structures are typically not easy to be observed. We will discuss them in this section.

The hump-shaped structure in Fig. 7(a) and the triangular bulge in Fig. 7(b) represent structures appearing before the hairpin vortex. These structures manifest as LSSs from the perspective of plane-view, such as the traditional hydrogen bubble time lines shown similar to Fig. 6. The lift-up and oscillation of LSSs will induce the high shear layer at its sides, where an initial roll-up occurs, as shown in Fig. 7(c). As mentioned in Section 3, the instability of LSSs will also produce a 3D inflectional profile, which will cause a lateral deformation of vortex lines. Also, they are considered as the initial stage of the hairpin vortex. This process has also been observed in previous investigations^{[1, 27, 37]}.

But what is the essence of the LSS? The model proposed by Smith roots the initiation of the LSS in the local adverse pressure gradient. However, what causes this local adverse pressure gradient is not clear. As pointed out by Lee^[16], a pair of oblique waves will produce a 3D wave structure which holds the feature of creating a region of low momentum from the wall, and perturbs the shear layer. This low-momentum region manifests as the LSS at the near-wall boundary layer. Thus, the aforementioned streak-induced scenario may be the same as the wave-induced scenario in essence. According to the results of the present study and also previous observations^{[16, 5]}, it is found that the LSS actually consists of 3D waves travelling downstream at approximately 65%U_∞ whose lift-up and amplification contribute to the burst. The results of applying the 3D LAVD method in the flow field of the initial roll-up stage reveal that vortices develop at the most inflectional regions, and once the vortices are present, there will be an interaction between waves and vortices. The complexity of the turbulent boundary layer lies in the evolution, symbiosis, and interaction between structures. The possible reason why wave structures are seldom observed will be illustrated below.

It is known that the LSS is mostly observed in the near-wall region and has never been reported as a vortex structure. Most researchers concentrate on the vortices interaction and their merging nearby, but ignore the low-momentum region itself. Thus, the internal 3D coherent structures of the LSS are easily to be ignored. Besides, hydrogen bubbles or other dye fluids on the front and rear of a single structure will come together with those of the adjacent structures, which leads to a superficial phenomenon of an integral low-speed region. Therefore, a great deal of caution should be paid in reading the record movies of hydrogen bubbles. Nevertheless, the visualization technique is still almost impossible to acquire 3D data for a structure at an instant time. Therefore, the time-resolved 3D particle image velocimetry (PIV) with sufficient space resolutions is strongly suggested to verify structures.

It is difficult to extract a real logical structure from the flow data in the absence of a good method for reconstructing the flow structure, especially when there are two or more structures in the boundary layer. When the structure has interactions, the Eulerian method may lose its power, and any structure derived from an iso-surface alone is often incomplete and sometimes misleading. Since the structure of an iso-surface is one thing while a real physical structure is another, it is necessary to establish a general method for data reconstruction. It is worth noting that most people concentrate their postprocessing on finding the structure with high vorticity. However, not all the coherent structures are vortices. In the wave structure, for example, there is little vorticity.

There are various definitions and detection methods of vortices, and most of them are frame-dependant based on the localized Eulerian criteria, from which reading the cause and effect is rather difficult. In contrast, the Lagrangian tracing method has shown its effectiveness in capturing structures in complex flows with the DNS^{[26-27, 69-72]}. Though, precisely extracting a structure in the experiments is just a beginning in the turbulent boundary layer. It needs time-resolved experimental techniques that have enough space resolution with low noise. The advantage of Lagrangian criteria is that extracting the evolution and interaction of structures is possible, which is usually absent in the previous investigation by applying the Eulerian method. Since the Eulerian method holds its superiority in instantaneous velocity fields especially where the viscous effect cannot be ignored, the combination of Lagrangian-based criteria with those traditional Eulerian methods is a potential path to obtain a complete structure as well as their evolution.

Overall, it must be admitted that the fact-based framework of hairpin-based structures is useful in understanding turbulent boundary layers. The present work just provides another perspective to view the early turbulent flow on a flat plate. Since the 3D wave structures could be traced back to early transitional boundary layers^[5], the generation of various coherent structures may be considered as the results of metamorphosis and evolution of 3D wave structures and their later symbiosis and interaction.

Many viewpoints about the hairpin vortex cannot be united due to the diversified observation in experiments and simulations. The hairpin-based framework is indeed one of the good depictions in wall-bounded turbulent flows, though the origin of hairpins vortex is in lack of physical and clear explanation. This paper tries to clarify the history and different explanations of the framework, and discusses the limitations in experimental techniques and postprocessing methods. Besides, new results based on the time-resolved Tomo-PIV are presented. The evolution of different coherent structures is extracted from the 3D velocity data sets by applying the Lagrangian tracking method. It is found that the LSS manifests as a 3D wave structure. The hump-shaped structure located in the streaky region will evolve into the triangular bulge whose edges oscillate violently due to the instability caused by fluid activity of lateral inrush and upward ejection. 3D wave structures reconstructed from the evolution of material surfaces seem to support the SCS theory^{[16, 5]}. The dimensions of the 3D wave are W⁺ ≈ 80, H⁺ ≈ 80, and L⁺ ≈ 200. The propagation speed of the wave is approximately 65% of the freestream velocity. By using the LAVD vortices detecting method, quasi-streamwise vortices are found at the sides of the 3D wave. According to the observation of the author and also the previous study, it is concluded that the 3D wave structure is the initiator of hairpin vortices. Due to the advantage in identifying coherent structures, the Lagrangian method seems to be a promising tool in resolving a large number of data derived from the DNS and Tomo-PIV.