1 Introduction

Turbulent flows exist in many circumstances,e.g.,chemical reaction,atmosphere,ocean, and turbulent thermal convection in the Sun. The turbulent flows constrained by the wall are also called wall-bounded turbulent flows. In wall-bounded turbulent flows,the coherent structures,which have spatial-temporal organized characteristics,contribute to the turbulent production significantly,and are the fundamental structures in many practical applications^{[1, 2, 3]}. In recent years,many efforts have been made,aiming to have a better understanding of the dynamics of these coherent structures,to establish reasonable turbulent models,or to develop the methodology of the turbulent manipulation^[4].

Falco^[5] found the large scale motion (LSM) in the turbulent boundary layer (TBL). Head and Bandyopadhyay^[6] pointed out that the LSM consisted of many hairpin vortexes,which formed a vortex packet. Moin and Kim^[7] reported a similar result with the direct numerical simulation (DNS) method. Meanwhile,many studies were devoted to revealing the topology of the coherent structures and their evolution mechanism. Robinson^[8] presented an idealized scheme for the distribution of the vortex structures in different regions of the TBL. Acarlar and Smith^[9] experimentally reproduced the hairpin vortex in a laminar boundary layer by using a hemisphere protuberance,and found the velocity profile,which eventually developed a remarkable similarity to the TBL. Adrian et al.^[10] proposed an idealized scheme of nested packets for the hairpin vortex growing up from the wall. Recently,with the DNS method,Wu and Moin^[11] found that a mass of symmetric hairpins in the serried ranks existed in a turbulent channel flow. They called these vortexes “hairpin forest”.

Alfonsi^[12] found that hairpin vortex provided an active mechanism for the formation of the streaks^{[13, 14]}. Tang et al.^[15] identified the meandering streaks in the log-low region,and found that the arrangement of these low/high-speed streaks staggered in the streamwise-spanwise plane tightly (see Fig. 9 in Ref. ^[15]). They claimed that the streaks in the log-law region were completely different from those in the near-wall region proposed by Kline et al.^[1]. The aim of this work is to experimentally study the detailed topological information around the local adjacent area of the low/high-speed streaks in the log-law region by using the tomographic time-resolved particle image velocimetry (TRPIV) technique.

The present paper is organized as follows. Section 2 describes the experimental setup and the basic flow field. Section 3 presents the method used for extracting the topological informa- tion. Section 4 discusses the connection between the spatial topological eigen-structures of the physical quantities and the low/high-speed streaks. Finally,the conclusions and remarks are given in Section 5.

2 Experiment setup

The tomographic particle image velocimetry (PIV) experiments are conducted in a water tunnel at Delft University of Technology by a team from German Aerospace Center,LaVision company,and Delft. The experimental dataset was first presented by Schroder et al.^[16],who gave a detail of the experiment with a description of the tomographic TRPIV. Here,we briefly recall the measurement setup and the dataset.

Figure 1 presents the schematic of the experimental arrangement. The plate dimension is 2 500mm×800mm. The measurement is made of a location of 2 090mm downstream of the leading edge. The corresponding measurement volume is 63mm×15mm×68mm in the streamwise,wall-normal,and spanwise directions,respectively. The TBL flow is established along a vertically mounted flat acrylic glass plate at a free-stream velocity of 0.53m·s−1. The TBL flow on the observation side is tripped by a spanwise attached zig-zag band,150mm downstream the leading edge. The thickness of the TBL is δ = 38mm at the measurement location 2 090mm (see Fig. 1). The Reynolds number based on the momentum thickness and free-stream velocity is Re_θ = 2 460. The skin friction velocity is estimated at u_τ = 0.021 9m·s−1, which corresponds to a friction coefficient cf = 0.003 45. The temperature is kept constant while the turbulent level of the free-stream velocity is below 0.5%.

The tomographic TRPIV system includes a diode pumped double high repetition rate laser with a maximum pulse energy of 25mJ at 1 kHz and six cameras with a resolution of 1 024×1 024 pixels in the full frame modus. The light is introduced,parallel to the plate and perpendicular to the mean flow direction. Sufficient light is scattered by the polyamide seeding particles with a mean diameter of 56 μm. This volume is illuminated at the frequency of 1 kHz. Five sequences of 2 040 image volumes are captured during a period of 2 s. A volume self-calibration with the particle image volume of 32 voxel×32 voxel×32 voxel and the overlap rate of 50% and the three-dimensional-three-component (3D-3C) velocity vector field with the article image volume of 32 voxel×32 voxel×32 voxel and the overlap rate of 75% are successively performed by the DaVis 7.3 software. A series of instantaneous 3D-3C velocity vectors are obtained,whose volumes are over a grid of 99×22×99 measurement points,locating every 0.687mm (about 15 wall units) in the streamwise,wall-normal,and spanwise directions for each 2ms,corresponding to a sampling frequency 500Hz. A comparison between the profiles of the tomographic PIV and a planar PIV measured in the same flow field confirms that the position of the measurement volumes is in the logarithmic layer,ranging approximately from y⁺ = 60 to y⁺ = 375 (in wall viscous units) in the wall-normal direction.

3 Digital image process

Multiscale,quasi-periodic,and space-time correlation are the basic characteristics of co- herent structures. The obtained instantaneous 3D-3C velocity vector fields are disordered and chaotic,in which the coherent structures are embedding. A reliable and efficient technique must be developed to recognize the coherent structures. In the past half century,several methods have been proposed,e.g.,the variable-interval time-averaging (VITA) scheme^[17],the quadrant splitting scheme^[18],the wavelet analysis^[19],and the linear stochastic estimation^[20]. Jiang et al.^[21] and Liu et al.^[22] proposed the locally-averaged-structure function (LASF) to describe the local deformation and relative motion of the coherent structures. Yang and Jiang^[23] extended the LASF to the 3D spatial flow field measured by the tomographic TRPIV,and identified the spatial topological mode of the coherent structures. The 3D version of the spatial LASF is defined as follows:

Equation (1) indicates the local stretch-compression (see Eq. (1a)),the shear of the local deformation (see Eq. (1b)),and the rotation of the local deformation (see Eq. (1c)) at different scales for the streamwise velocity components of the turbulent flows. Here,u stands for the streamwise velocity component,and x₀ stands for the central position of the multiscale defor- mation in the streamwise direction. u(x,y,z) is the streamwise velocity with the scale of 2ℓ and the center locating at x₀. The structure function defined here acts as a low-pass filter^[24]. With this method,the disorder part will be removed,and the coherent part will be preserved. Therefore,the spatial coherent structure can be recognized properly.

Similarly,the spatial LASF for the spanwise velocity v and the wall-normal velocity w are defined as follows:

and

In the local streamwise adjacent areas of the low/high streaks,the streamwise fluctuating velocities of the low/high-speed streaks have opposite signs. It means that the local stretch- compressive deformation exists in these regions. A detection function is then designed as follows:

where H stands for the structure that the high-speed fluid is at the center and the low-speed fluids are at the upstream and downstream,and L stands for the structure that the low-speed fluids are at the center and the high-speed fluid is at the upstream and downstream. Therefore, D(ℓ,b) = 1 indicates that the fluids at the upstream are stretched,and the low-speed fluids in the downstream are compressed. D(ℓ,b) = −1 implies that the fluids at the upstream are compressed,and the high-speed fluids at the downstream are stretched and swept. A phase- lock-average technique is then adopted to extract the spatial topological mode of the coherent structures,i.e., where h i represents the ensemble average operator,f(ℓ_j,x) is the physical variable such as the fluctuating velocity or vorticity,Nj is the sample number of the H or L structures with the jth scale,and ℓ_j represents the spatial jth scale length.

4 Results and discussion

The spatial LASFs in four scales are calculated according to Eq. (1a) by using the dataset of time series 3D-3C velocity vector fields. Then,the H events are detected by the sampling criterion of Eq. (4) with the fourth scale,and the eligible data points are abundant for the analysis. To obtain the spatial phase average topological information around the H events, the same number of data volumes is selected for the phase alignment-superimposed average according to Eq. (5). The spatial size of each volume is

grids,corresponding to wall viscous units. The spatial detected eligible point is at the bottom center of each selected data volume. We analyze the results of the spatial topological mode of the coherent structure. All the detected points are on a horizontal plane at in the log-law region. The physical quantities involved in the spatial topological mode include the fluctuating velocity (u′,v′,w′) and the vorticity (ω₁,ω₂,ω₃).

Figure 2 shows the 3D topological contour modes of the fluctuation velocity field around the testing center of the ejection. The coordinates of the eligible point in the volume are

Figures 2(a),2(c),and 2(e) show the 3D topological contour modes of the streamwise fluctu- ating velocity u′,the wall-normal fluctuating velocity w′,and the spanwise fluctuating velocity v′,respectively. Figures 2(b),2(d),and 2(f) present the cross-sections of the local fluctuation velocity. As shown in Fig. 2(a),Regions A,B,and C stand for the high-speed fluids,while Regions D,E,and F represent the low-speed fluids. It is observed that the high-speed streak at the upstream (see Region A) is surrounded by two upstream low-speed streaks (see Regions E and F) and a larger downstream low-speed streak (see Region D). According to the 3D topol- ogy of the wall-normal fluctuating velocity in Fig. 2(c),the blue region (the deep-colored area in the middle of the lower part of the deep-colored ring) with negative w′ corresponds to the areas with high-speed fluids,while the yellow region (the deep-colored area in the middle of the upper part of the deep-colored ring) with positive w′ corresponds to the areas with low-speed fluids. In order to achieve a more pronounced effect,the contour plots of u′ and w′ with the tangent velocity vector are displayed in Figs. 2(b) and 2(d),respectively. The cross-section is on the plane of z⁺ = 0. The high-speed fluid (the red region (the deep-colored area on the right-side) in Fig. 2(b)) has a negative value of w′ (see the blue region (the deep-colored area on the right-side) in Fig. 2(d)),while the low-speed fluid (the blue region (the deep-colored area on the left-side) in Fig. 2(b)) has a positive value of w′ (see the red region (the deep-colored area on the left-side) in Fig. 2(d)). It indicates that the high-speed fluid sweeps down towards the wall,and the low-speed fluid ejects up against the wall. On the plane of z⁺ = 0 (see Fig. 2(b)), we have already acknowledged that high-speed fluids (see Region A in Fig. 2(a)) and low-speed fluids (see Region D in Fig. 2(a)) appear one after another along the free-stream direction. The downstream low-speed fluid (see Region D in Fig. 2(a)) is at the center of the spanwise direction. According to Figs. 2(e) and 2(f),we can see that the distribution of the spanwise fluctuating velocity v′ reveals a quadrupole or four-leaf clover mode. From the location perspective,it is not difficult to draw that the spanwise fluctuating velocity v′ is opposite on both sides of the low-speed streak at the downstream center. Furthermore,it is contrary to the status at the upstream.

Figure 3 shows the 3D topological modes of the streamwise vorticity ω₁ (see Fig. 3(a)) and the wall-normal vorticity ω₃ (see Fig. 3(c)). The relative positions of the contours of ω₁ and ω₃ and those streak structures (see Fig. 2(a)) are also displayed in Fig. 3(b) and Fig. 3(d), respectively. From the two-level contours of ω₁ in Fig. 3(a),the quadrupole or four-leaf clover mode is revealed. There are two pairs of dipole vortex packets (see Pairs A and B) with opposite values of ω₁. From Fig. 3(b),we can also find that there are two pairs of antisymmetric stream vortexes in the local adjacent region of the high/low-speed streaks. Pair A generates the low- speed streak (see Region D in Fig. 3(b)),and Pair B has a close relation with the high-speed streak (see Region A in Fig. 3(b)).

In the two-level contours of ω₃ (see Fig. 3(c)),a sextupole packet which contains three pairs of regions (denoted as Pair 1,Pair 2,and Pair 3) is recognized. The values of each pair of wall-normal vortexes are opposite. It indicates that each pair of the antisymmetric wall-normal vortexes corresponds to a low-speed streak nipped in the middle.

These topological modes of fluctuating velocities and vorticities around the local adjacent region of the low/high-speed streaks offer us valuable clues for seeking the connection between the vortexes and the streaks. We summarize these essential features as follows. The coalescence of two low-speed streaks at the upstream generates the larger scale low-speed streak at the downstream. Sweep and ejection events occur in succession along the streamwise direction, corresponding to a high-speed streak and a low-speed streak,respectively. At the bottom of the measurement volume,fluids gather around the low-speed streak,while spread out near the high-speed streak. From the topological modes of ω₁ and ω₃,it is found that except for the two missing parts (indicated by the two dashed red circles in Fig. 3(a)),the other regions correspond to each other. This indicates that a three-pair hairpin vortex exists in the adjacent region. Moreover,there is an association among the quadrupole structure of ω₁,the sextupole structure of ω₃,and the low/high-speed streaks,and the three pairs of inclined vortexes are closely tied to the streaks.

According to Robinson^[2],Acarlar and Smith^[9],and Adrian et al.^[10],the low-speed streak lies in the middle of the two legs of the hairpin vortex packets in the streamwise direction. We thus deem here that a three-pair hairpin vortex bestrides the three low-speed streaks. The term hairpin may present for different forms,such as cane,hairpin,horseshoe,and omega shaped. Here,we choose the classic hairpin shape as the typical vortex mode. Other forms cannot be excluded in the log-law region of the TBL.

Based on the feature and presumptive form of the vortex mentioned above,we propose a model here to explain these topological results in Fig. 4. In this ideal 3D topological model of the relationship between the vortex and streaks in the adjacent region of the low/high-speed streaks,the regions correspond to the regions in Fig. 2(a). The framework is a representative of the selected volume. Taking the three pairs of wall-normal vorticities (see Fig. 3(c)) as the prototype,and considering the streamwise component of the vortexes,we can obtain the brown regions (Pairs 1,2,and 3) for the three pairs of hairpin vortexes and that each region concludes two counter-rotating vortex legs. For the adjacent region,we neither depict the other hairpin vortexes from the same hairpin packet along the low-speed streaks,nor state the length of these streaks. The two legs of the hairpin vortex (Pair 1) at the downstream generate the low-speed fluid which ejects to the upstream. The two legs,which consist of the left leg of Pair 2 and the right leg of Pair 3,produce the high-speed fluid sweeping towards the wall. At the bottom of the selected volume,the distribution of the spanwise velocity of the fluids also agrees with that shown in Fig. 2(f). The present model proves its credibility by the logical bridge between the experimental results and the conjectures.

5 Conclusions

In summary,the tomographic TRPIV technique is used to investigate the coherent structures in the adjacent region of the low/high-speed streaks along the streamwise in the TBL with the Reynolds number of about 2 460. A spatial LASF and a detection function D(ℓ,b) are proposed to identify the coherent structures. According to the results of the topological analysis on the velocity and vorticity at y⁺ = 120,we have the following conclusions:

(i) The spatial LASF can describe well the spatial deformation and rotation of the multiscale coherent structures in wall-bounded turbulent flow.

(ii) Ejection events are closely linked to low-speed fluids,while sweep events are closely linked to high-speed fluids. Moreover,in the adjacent region of the low/high-speed streaks, ejection and sweep events occur in pairs along the free-stream direction.

(iii) A quadrupole structure and a sextupole structure are detected for the first time by the experimental method. The topological information reveals the existence of a three-pair counter-rotating hairpin vortex in the adjacent region of the low/high-speed streaks.

(iv) According to the analysis results we have obtained,a 3D model is proposed by using the typical structure of the hairpin vortex packet. The 3D model can answer the connection among the low/high-speed streaks,the ejection/sweep event,and the hairpin vortex. Therefore,the packet,which consists of three hairpin vortexes,plays a key role in the adjacent region of the low/high-speed streaks.

We provide some comments on the present work. Due to the limitation of the tomographic TRPIV technique,the thickness of the measurement volume and the resolution are still not enough in the wall-normal direction. Moreover,a long time recording (e.g.,few minutes with full resolution) is necessary for the study of the temporal and spatial evolutions of these coher- ent structures. However,due to the limitation of the memory of the measurement system,it is not available currently. We hope that with the development of the measurement and image processing technique,a long time record with resolution up to the Kolmogrov scale can be available in the near future. It will help us understand more details in wall-bounded turbulent flows,e.g.,the momentum balance in which the inverse cascade is relevant^[3]. Acknowledgements The authors would like to gratefully acknowledge the German Aerospace Center Institute of Aerodynamics and Flow Technology at Gottingen for providing the tomographic TRPIV dataset of the TBL.