Science in China Series G: Physics, Mechanics & Astronomy © 2009

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Identification of Lagrangian coherent structures in the turbulent boundary layer PAN Chong, WANG JinJun† & ZHANG Cao Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

Using Finite-Time Lyapunov Exponents (FTLE) method, Lagrangian coherent structures (LCSs) in a fully developed flat-plate turbulent boundary layer are successfully identified from a two-dimensional (2D) velocity field obtained by time-resolved 2D PIV measurement. The typical LCSs in the turbulent boundary layer are hairpin-like structures, which are characterized as legs of quasi-streamwise vortices extending deep into the near wall region with an inclination angle θ to the wall, and heads of the transverse vortex tube located in the outer region. Statistical analysis on the characteristic shape of typical LCS reveals that the probability density distribution of θ accords well with t-distribution in the near wall region, but presents a bimodal distribution with two peaks in the outer region, corresponding to the hairpin head and the hairpin neck, respectively. Spatial correlation analysis of FTLE field is implemented to get the ensemble-averaged inclination angle θ R of typical LCS. θ R first increases and then decreases along the wall-normal direction, similar to that of the mean value of θ. Moreover, the most probable value of θ saturates at y+=100 with the maximum value of about 24°, suggesting that the most likely position where hairpins transit from the neck to the head is located around y+=100. The ensemble-averaged convection velocity Uc of typical LCS is finally calculated from temporal-spatial correlation analysis of FTLE field. It is found that the wall-normal profile of the convection velocity Uc(y) accords well with the local mean velocity profile U(y) beyond the buffer layer, evidencing that the downstream convection of hairpins determines the transportation properties of the turbulent boundary layer in the log-region and beyond. turbulent boundary layer, coherent structures, vortex identification scheme, finite-time Lyapunov exponents, hairpin-like vortices

Turbulence, existing universally in nature and engineering, has been regarded as one of the most complicated problems in fluid dynamics. Since Kline et al.[1] revealed the existence of streaky structures and burst events in the turbulent boundary layer in the 1960s, numerous efforts have been devoted to such coherent structures/motions. Now, it is well accepted that coherent structures/motions with different shape and scale constitute the self-sustaining mechanisms of wall turbulence. Moreover, vortices are regarded as the dominant structures, which play important roles in the generation of other coherent － structures/motions, e.g., streaks, burst and bulges[2 4]. Hairpin-like vortices, large-scale transverse vortices and streamwise vortices have all been observed in wall tur-

bulence, among which hairpin-like vortices are regarded as the fundamental elements, whose reproduction and evolution are the prerequisite for the self-sustaining of other coherent structures/motions[5]. Hairpin-like vortices in wall turbulence are immersed in chaotic background fluctuations with small scale and significant intensity, and their symmetry and integrity are apt to be destroyed during downstream evolution. In this sense, hairpin-like vortices with satisfying symmetry and integrity are seldom observed in the turbulent Received May 22, 2008; accepted September 1, 2008 doi: 10.1007/s11433-009-0033-1 † Corresponding author (email: [email protected]) Supported by the National Natural Science Foundation of China (Grant Nos. 10425207 and 10832001)

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boundary layer. Instead, those structures with transverse-rotated heads and streamwise-elongated legs, such as arch-like vortices and ring-like vortices, are regarded as generalized hairpins[6]. To study their dynamical characteristics, hairpins were first introduced into the laminar boundary layer in － experiments[7 9]. These manually excited hairpins have satisfying symmetry and integrity. Moreover, the laminar background ensures the feasibility of identifying these hairpins and tracking their downstream evolution with less difficulty by visualization techniques. On the other hand, the advances in the technique of vortex identification schemes enable us to identify vortical structures from turbulent velocity field in a quantitative way. The widely-used vortex identification schemes include Q-criteria[10], λ2-criteria[11], Δ-criteria[12] and λci-criteria[13]. All these schemes are based on Eulerian system, using certain characteristics of invariant of velocity gradient tensor ∇v to depict vortex. In order to compute eigenvalues of ∇v , three-dimensional instant velocity field with adequate spatial resolution is required, which is difficult for present experimental techniques. Therefore, these Eulerian-based schemes are now mainly used to identify vortical structures from direct numerical simulation (DNS) database of turbulence channel flow[6,13]. Another kind of schemes belongs to Lagrangian system. Instead of invariant of ∇v , time-integrated properties of fluid particle trajectories are used to identify vortical structures. One of the typical Lagrangian-based schemes is Finite-Time Lyapunov Exponents (FTLE) method. Green et al.[14] educed hairpin-like Lagrangian structures from DNS database of turbulence channel flow. However, no studies on identifying Lagrangian structures from experimentally-measured wall turbulence have been reported. Taking this as the primary goal of the present work, we measure 2D velocity field of a flat-plate turbulent boundary layer by time-resolved particle image velocimetry (PIV), and apply the FTLE method to educe Lagrangian coherent structures (LCSs) in the turbulent boundary layer. Moreover, the phenomenological characteristics of LCSs and their convection statistics are further analyzed.

chaos. It measures the averaged expansion rate of two neighboring trajectories in phase space. For time-dependent dynamical systems, Lyapunov exponent is usually integrated within a finite time interval, the integrating direction along time dimension may be either forward or backward. In fluid dynamics, the positive value of forward-integrated FTLE at a certain space point reveals that the trajectories of two neighboring fluid particles initialized at that point with an infinite distance gradually separate from each other, and the relative distance between these two particles increases exponentially with time. From Lagrangian viewpoint, vortex boundary may be regarded as the hyperbolic material surface which distinguishes fluids inside the vortex from the surrounding background. Considering a pair of particles with an infinite distance which are separated by the vortex boundary, due to the rotating/spiraling motion of the vortex, these two particles either depart from or approach each other within a short time interval. The divergence/convergence rate is inversely proportional to their initial distance to the vortex boundary, thus leading to the local maximum/minimum of FTLE at the vortex boundary. Haller et al.[15,16] mathematically proved that particle pairs located at the vortex boundary own the maximum stretching/folding rate. In this sense, the vortex boundary can be objectively defined as the ridges in FTLE field which are gradient lines of FTLE field transverse to the direction of minimum curvature. Physically speaking, FTLE field may be computed by: (i) evenly distributing imaginary particles with sufficient number in the flow field at initial time t0; (ii) tracking the convection of both the particle located at the specified space point Xi,j and its nearest neighbors with initial distance small enough over time interval T; (iii) calculating the relative distance D(T; t0; Xi,j) between Xi,j and its neighbors at the time of t0+T. To advect particles in the flow field, Eulerian velocity field with sufficient temporal resolution is needed. Figure 1 gives the illustration of advecting particle groups. The relative distance between the central particle Xi,j and its neighbors is defined as 2 1 D 2 (T ; t0 ) = X i , j (t0 + T ) − X i + m, j + n (t0 + T ) . ∑ N m =±1, n =±1

1 Finite-Time Lyapunov Exponents method

(1) Haller et al. [16] derived that the expansion rate σ T ( x0 , t0 ) of the relative distance between neighboring

Lyapunov exponent is a fundamental parameter to depict

trajectories is equal to the square of the largest singular

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249

Figure 1 Illustration of advecting particle groups in Eulerian velocity field during calculating FTLE.

value of the deformation gradient ∂x(t0 + T , x0 , t0 ) ∂x 0 :

σ T ( x0 , t0 ) ⎛ ⎡ ∂x(t + T , x , t ) ⎤T ⎡ ∂x(t + T , x , t ) ⎤ ⎞ 0 0 0 0 0 0 ⎜ = λmax ⎢ ⎥ ⎢ ⎥ ⎟ , (2) ⎜⎣ ∂x0 ∂ x 0 ⎦ ⎣ ⎦ ⎟⎠ ⎝ where x(t0+T, x0, t0) is the spatial position of the particle initiated at (x0, t0) after advecting over time interval T. The corresponding FTLE at point x0 is then calculated as 1 (3) FTLET ( x0 , t0 ) = log σ T ( x0 , t0 ). 2T For a given integration time T, FTLE varies as a function of space x0 and initial time t0. The ridges of FTLE field with fixed t0 depict regions of localized maximum material stretching, and thus may be regarded as vortical structures. For the case of integrating trajectories in backward time direction (T<0), the ridges of FTLE field indicate attracting material lines with maximum convergence rate, analogous to the accumulation of dye/ hydrogen bubbles collected by vortical structures in flow visualization. Therefore, the so-depicted structures are essentially attracting Lagrangian coherent structures (LCS). On the contrary, the ridges of FTLE field with forward time integration (T>0) depict the material lines with localized maximum of divergence rate, thus representing the repelling LCS. Shadden et al.[17,18] studied fluid transport characteristics in vortex ring flows using the Lagrangian-based vortex identification method. Mathur et al.[19] identified the Lagrangian skeleton of quasi-2D turbulence. Green et al.[14] further educed hairpin-like Lagrangian structures from DNS database of turbulence channel flow. All these studies demonstrated that the FTLE method is superior to the traditional Eulerian-based vortex identification schemes in the following aspects. First, the FTLE 250

method objectively identifies coherent structures by ridges of FTLE field, while in Eulerian-based schemes the boundaries of vortical structures depend on manually-defined threshold. Second, velocity gradient tensor is no longer used as criteria function, thus relaxing the restriction on the spatial resolution of velocity field. Finally, the time-integrated arithmetic is insensitive to short-term imperfection of velocity data, and the section shape of 3D structures may be identified from time-resolved 2D velocity field when using Taylor’s frozen hypothesis. One of the shortcomings of this method is the computational cost: Since the FTLE calculation involves time-integration of particle trajectories from each point for each instant, this method is more time-consuming than any of the Eulerian schemes. Moreover, sufficient temporal resolution of velocity field is desired for the trajectory integration, thus increasing the difficulty in the experiment/simulation.

2 Experimental model and facility A flat-plate turbulent boundary layer was experimentally measured in a low-speed water channel in Beijing University of Aeronautics and Astronautics. The working section has a size of 600 mm×600 mm×3000 mm (height×width×length). The free-stream velocity is U∞ = 86.1 mm/s, and the free-stream turbulence intensity Tu is less than 0.8%. The boundary layer is developed on the surface of a plexiglas flat plate with a 4:1 elliptical leading edge. The flat plate, with a size of 10 mm×600 mm×2000 mm (thick×width×length), is horizontally positioned 200 mm above the bottom of the water channel. The angle of attack to the free stream is carefully adjusted so that flow around the elliptical leading edge has no separation. Transition is tripped by a 6-mm-indiameter rod transversely stuck to the wall at x=70 mm downstream from the leading edge, resulting in a fully developed turbulent boundary layer after x=700 mm. Time-resolved 2D PIV is used to measure the velocity field in the (x, y) plane (side-view plane) of the turbulent boundary layer. Seeding particles with median diameter of 5 μm and density of 1.05 g/mm3 are illuminated by a continuous laser sheet with thickness of about 1 mm and energy output of 1.5 W. A high-speed CCD camera (640×480 pixels) captures the sheet-sliced flow field with the maximum sampling frequency of 200 Hz. For a low-speed flow field (U<15 mm/s), the relative displacement of particle groups between two neighboring

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images is less than 5 pixels at the sampling frequency of 120 Hz. In this sense, the traditional PIV configuration of “double-pulsed laser and low-speed camera” may be replaced by the present combination of “continuous laser and high-speed camera”, which enables continuous measurement at sufficiently high spatial resolution. To conduct the side-view measurement, the laser sheet is positioned perpendicular to the flat plate along the symmetrical plane at x=1000 mm downstream from the leading edge. Special measures are taken to ensure the perpendicularity of the laser sheet and the optical axis of the CCD camera. Both the optical configuration and the coordinate definition are illustrated in Figure 2. The sampling frequency of the CCD camera is 200 Hz, with exposure time of 1 ms. The field of view is about 80 mm×60 mm (streamwise length×wall-normal width). Particle images are analyzed by MircroVec® with interrogation windows of 16×8 pixels and overlap rate of 50%. This software package adopts the approach of multi-grid iteration with window deformation[20]; the system error is thus reduced to be about 1%. Since the mean diameter of the illuminated particles dp is about 2 pixels, the phase lock effect is neglectable. Four replicates of image sampling are conducted, each capturing 5000 frames, corresponding to a time duration of 25 s. Figure 3 presents the dependency of the maximum velocity fluctuation intensity urms,max on the sample size N at x=1050 mm, y=3.41 mm. It is shown that urms,max reaches the convergence when N>18000, demonstrating that the sample size adopted in the present measurement is sufficient for the statistical analysis of the turbulent boundary layer. The software package MANGEN is used to calculate

Figure 3 Dependency of the maximum velocity fluctuation intensity urms,max on the sample size N in PIV measurement (x=1050 mm, y=3.41 mm).

FTLE field. The fluid trajectories are obtained by advecting particles in PIV-measured velocity field with a 4th-order Runge-Kutta-Fehlberg integration algorithm. Because of the discrete velocity data, the 3rd-order interpolation method is used to provide velocity information for any required spatial position.

3 Base flow Figure 4 shows the wall-normal profile of both the mean velocity U+(y+) and the velocity fluctuation intensity urms(y+) and vrms(y+) at x=1020 mm, 1035 mm and 1050 mm, in which urms and vrms are non-dimensionalized by free-stream velocity U∞, while U and y are non-dimensionalized by friction velocity uτ and inner scale y*, respectively. As shown in Figure 4(a), U+(y+) agrees well with Spalding formula[21]: ⎡ + y + = U + + e − KB ⎢e KU − 1 − KU + ⎣ (4) + 2 ( KU ) ( KU + )3 ⎤ − − ⎥, 2! 3! ⎦ where U+=U/uτ, y+=y/y*=yuτ/υ, K=0.4 and B=5.5. Figure 4(b) further shows that the urms peak is located around y+=14 with the peak value of about 0.15U∞, consistent with previous studies[22]. Moreover, the profiles of U+(y+), urms(y+) and vrms(y+) at three streamwise positions respectively collapse into one curve. This self- similarity evidences the setting up of the self-sustaining mechanisms of the fully developed turbulent boundary layer. The characteristic parameters of the present turbulent boundary layer at x=1050 mm are given in Table 1. Table 1 Characteristic parameters of the turbulent boundary layer at x=1050 mm

Figure 2 The optical configuration and the coordinate definition in the side-view PIV measurement.

U∞ (mm/s) δ (mm) 86.1 39.3

δ1 (mm)

θ (mm)

8.7

5.9

H Reθ 1.47 481

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uτ (mm/s) δ + 4.6461 173 251

Figure 4

Wall-normal profile of the mean velocity U+( y+) and the velocity fluctuation intensity urms( y+) and vrms( y+) in the turbulent boundary layer.

4 Typical LCS The present investigation mainly focuses onto attracting LCS identified by the ridges of backward-integrated FTLE field with integration time T = −2 s. Figure 5 illustrates a snapshot of FTLE distribution in (x, y) plane at certain instant, where only the value of FTLE larger than zero is shown. Two attracting LCSs in the turbulent boundary layer are visualized by FTLE ridges, both presenting an inclined streaky pattern with the upstream portion extending into the near-wall region and the downstream portion rising up into the outer region. Such structures are commonly visualized in FTLE field, thus representing the typical LCS in the side-view plane of the turbulent boundary layer. Green et al.[14] used the FTLE method to identify an isolated hairpin vortex generated by linear stochastic estimation in numerically simulated turbulence channel flow. The side-view shape of the hairpin vortex is similar to the present LCS in Figure 5. Figure 6 further shows a snapshot of FTLE distribution in (t, y) plane at x=1100 mm. Inclined LCSs depicted by FTLE ridges are frequently visualized, indicating “frozen” structures with streaky pattern successively passing through the vertical line at x=1100 mm. Most of these LCSs inflect significantly towards the wall at the downstream end, even forming half-closed ring-like structures for some circumstance. Combining Figures 5 and 6, it may be concluded that these typical LCSs are actually the projection of generalized hairpin vortices in the turbulent boundary layer onto the sideview plane. The legs of these hairpins retard in the nearwall region and elongate along the streamwise direction 252

due to mean shearing; while their heads rise up into the outer layer due to lift-up effect, sometimes reaching the height of y+=130. Moreover, Figure 6 shows three neighboring hairpins in the time range of t－t+Δt. These hairpins equivalently align in temporal dimension with successively lowering wall-normal position. The legs of the former hairpins are always connected with the heads of the latter, suggesting the parent-offspring relationship between these hairpins. Therefore, these hairpins can be regarded as one hairpin packet, which are formed through regeneration and self-organization mechanisms[9].

Figure 5 A snapshot of FTLE distribution in the (x, y) plane inside the turbulent boundary layer and the definition of LCS inclination angle θ. Only FTLE larger than zero is shown in the contour map. The black line represents the ridges of FTLE. The origin of the x axis is located at x=1050 mm.

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In their study on hairpin vortices in wall turbulence, Adrian et al.[23] found that once viewed in a reference frame moving at specified speed with main flow, the instant velocity vector in (x, y) plane may present circular or elliptical pattern with concentrated vorticity at some place, indicating the existence of spanwise vortices rotating around z axis; the convection velocity of these vortices is equal to the moving speed of the reference frame. Based on this observation, they identified hairpin vortices by the signature of spanwise vortex core located above a region of strong second-quadrant fluctuations (u<0 and v>0), which inclines at 30°－60° to the wall. Figure 7(a) and (b) compare this vortex identification criterion with the present FTLE method. The colormap in Figures 7(a) and (b) illustrates an instant distribution of FTLE field in (x, y) plane, and arrows represent the velocity vector in a downstream-moving frame with the moving speed Uframe of 0.7U∞ in Figure 7(a) and 0.9U∞ in Figure 7(b). The FTLE ridges in Figures 7(a) and (b) indicate two hairpin vortices with different scale and position: The downstream hairpin is fully grown while the upstream one is newly born from near-wall erupted fluids; the wall-normal position of their heads is respectively located around y+=90 and 30. In the dotted rectangle in Figures 7(a) and (b), the instant velocity vector presents a circularly closed pattern with the circular center located at the downstream end of one LCS, indicating its spanwise rotating motion. Noting that the criterion of “circularly closed relative velocity vector” only identifies spanwise vortices with convection velocity equivalent to the moving speed of the reference frame, the relative velocity vector in Figures 7(a) and (b) cannot reveal the head of another hairpin outside the dotted rectangle. If the exact position of the vortex center is unknown in advance, its convection velocity cannot be easily estimated. Therefore, an iteration procedure is needed when applying the velocity vector method. For the present case, Uframe has been iterated in the range from 0.6U∞ to 1.0U∞ with a spacing of 0.05U∞ until the values of 0.7U∞ and 0.9U∞ are determined as the most proper one to reveal the rotating pattern of the hairpin heads. As shown in sec. 5, the values of 0.7U∞ and 0.9U∞ are equivalent to the ensemble-averaged convection velocity of typical LCS at y+=30 and 90. Comparing to the limitation of both the subjective estimation of the convection velocity and the disability of simultaneously identifying all structures in the velocity

vector method, the FTLE method may objectively reveal the spatial shape of LCS without any manually selected threshold. Moreover, all structures with different shape, position and convection velocity can be simultaneously identified, thus enhancing the efficiency. Figure 7(c) further presents the distribution of instant spanwise vorticity ωz by colormap. Combining with Figures 7(a) and (b), we find that the upstream new-born hairpin owns concentrated negative vorticity with irregular shape around the hairpin head. However, no significant vorticity concentration is observed around the head of the downstream full-grown hairpin. This again demonstrates the idea that the criterion of vorticity concentration itself is insufficient for revealing vortical structures in shear flow.

5 Statistical characteristics of typical LCS As shown in sec. 4, typical LCS in the turbulent boundary layer presents inclined streaky pattern extending into the near-wall region with inclination angle θ to the wall. Being an important parameter to measure the spatial shape of LCS, the statistics of θ whose definition is illustrated in Figure 5 is analyzed in this section. LCS in (x, y) plane may be extracted from instant FTLE field by the following procedures: (i) searching spatial points in each instant FTLE field where the spatial derivative

( ∂FTLE

∂x ) + ( ∂FTLE ∂y ) 2

2

approaches zero and the

corresponding FTLE is larger than zero; (ii) tagging a point found in (i) as localized peak if it owns the maximum value of FTLE in its 5×5 neighborhood; (iii) connecting all of the localized peaks to form ridges of the FTLE field; (iv) only those ridges with length larger than specified threshold being regarded as typical LCS, corresponding to the projection of generalized hairpins onto (x, y) plane. For each LCS subtracted through the above procedures, its inclination angle θ at different wall-normal positions is then calculated. Figure 8 presents the probability density distribution of LCS inclination angle P(θ ) at three wall-normal positions, y+=15, 55 and 115. In the near-wall region of y+=15 and 55, P(θ ) accords well with t-distribution: ⎛ υ + 1 ⎞ ⎡υ + ⎛ θ − μ ⎞ ⎤ Γ⎜ ⎟ ⎢ ⎜ ⎟⎥ ⎝ 2 ⎠ ⎢ ⎝ σ ⎠⎥ P (θ ) = υ ⎛υ ⎞ ⎥ σ υ πΓ ⎜ ⎟ ⎢⎢ ⎥ ⎝2⎠⎣ ⎦

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−

υ +1 2

,

(5)

253

Figure 6

A snapshot of FTLE distribution in the (t, y) plane inside the turbulent boundary layer (x=1100 mm, Δt=0.5 s).

Figure 7 Comparison of the capability of three vortex identification techniques, the FTLE method, the velocity vector method and the vorticity method. (a) and (b) colormaps show the distribution of the FTLE field and arrows show the relative velocity vector in the moving reference frame, wherein the right column is the enlarged view of the dotted region in the left column; (c) colormap shows the distribution of spanwise vorticity field ωz and arrows show the relative velocity vector. The moving speed of the reference frame is Uframe=0.7U∞ in (a) and (c), while Uframe= 0.9U∞ in (b), and the origin of the x axis is located at x=1050 mm.

254

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Figure 8 Probability density distribution of inclination angle θ of typical LCS at three different height. (a) y+=15; (b) y+=55; (c) y+=115.

where μ is the mean value of θ, σ is the corresponding standard deviation and Γ(x) is Gamma function. Distinguishing from the near-wall case, P(θ ) in the outer region of y+=115 exhibits a bimodal distribution with two peaks around 0° and 25°. As shown in Figures 6 and 7, both the head and the neck of instant LCS frequently appear around y+=115. Therefore, the most probable peak around θ = 25° may be associated with the kinked hairpin neck, while another peak around θ = 0° may be related to hairpin heads which always inflect towards the wall. Besides calculated from individual LCS, the characteristics of LCS inclination angle may be also obtained from spatial correlation analysis of FTLE field. Twopoint correlation coefficient of time-series of FTLE is defines as follows: RFTLE,FTLE ( x0 , y0 , Δx, Δy ) =

FTLE A ( x0 , y0 )FTLE B ( x0 + Δx, y0 + Δy )

σ Aσ B

,

(6)

where the reference point A is fixed at (x0, y0) and the moving point B offsets from A with relative distance of Δx and Δy. Figure 9(a) presents four sets of RFTLE, FTLE(x0, y0, Δx, Δy) contour in (Δx, y) plane. The reference points (point A) are located at the same streamwise position (x0=1050 mm), but their wall-normal position varies as y0+=15, 50, 85 and 120. As shown in Figure 9(a), all these RFTLE, FTLE contours present inclined streaky pattern, similar to that of instant LCS discussed in the previous section. Such characteristic topology may be identified by ridges in the RFTLE, FTLE field. Since RFTLE, FTLE measures the similarity degree between time variations of FTLE at different spatial points, and FTLE quantitatively depicts the signature of instant LCS, the ridges in the RFTLE, FTLE field may be regarded as the finite portion of ensemble-averaged LCS around the reference point (x0, y0). On considering the x-independency of the statis-

tical prosperities of the present turbulent boundary layer, the inclination angle of typical LCS may be only a function of y position. Figure 9(b) presents the wall-normal distribution of θR calculated by the spatial correlation method, together with the mean and the most probable value of θ calculated from individual LCS. Generally speaking, the profile of the mean value of θ is close to that of θR within a large wall-normal range, evidencing the conjecture that θR of RFTLE, FTLE ridge may indicate the local inclination angle of ensemble-averaged typical LCS. The wall-normal variation of these two parameters may be well predicted by a second-order polynomial fit, characterized as an initial increase and a following decrease after y+=90. The most probable value of θ agrees with both the mean value of θ and θR below y+=60. This is consistent with Figures 8(a) and (b), where P(θ ) in the near-wall region presents the concentrated peak with a small width, indicating the uniform shape of typical LCS in this region. The deviation of the most probable value of θ from both the mean value of θ and θR occurs in the outer region: The most probable value of θ increases gradually, soon exceeding its counterparts beyond y+=60. Saturation reaches at about y+=100 with the value of about 24°, and then the most probable value of θ also presents a continuous decrease, consistent with the observation that instant LCS inflects towards the wall in the outer region. Therefore, it is inferred that the most likely position where hairpins transit from neck to head is located around y+=100. Time variation of FTLE at fixed point contains the information of the passage of typical LCS with quasi-periodicity; in this sense, their downstream convection velocity can be calculated by temporal-spatial correlation analysis of FTLE field. The temporal-spatial correlation coefficient of FTLE between two points A

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255

Figure 9 Calculation of inclination angle θR of time-average typical LCS using the spatial correlation method. (a) Variation of spatial correlation of LCS signature RFLTE,FTLE(x0,y0,Δx+,Δy+) along the wall-normal direction. Contour with the solid line: y0+=15; with the dashed line: y0+=50; with the dash-dotted line: y0+=85; with the dotted line: y0+=120. The contour level for RFLTE, FTLE ranges from 0.4 to 0.8 with spacing of 0.2. In each contour, the solid-dotted line is the ridge which connects the points owning the localized peak value along the y direction, and thus depicts the most-probable portion of LCS. (b) Wall-normal distribution of the inclination angle of LCS, ○: θR; the solid line, the 2nd-order polynomial fit; Δ, the mean value of θ; ∗, the most probable value of θ.

Figure 10 Calculation of convection velocity using the temporal-spatial correlation method: (a) Illustration of spatial-temporal correlation of LCS signature RFLTE,FTLE and the definition of convective time delay τc, x0=1050 mm, Δx=10 mm, y0+=50; (b) the wall-normal profile of convection velocity Uc(y+) of typical LCS (indicated by circle). The solid line is the local mean velocity profile U(y+) in the turbulent boundary layer.

and B with specified time delay τ is defined as follows: RFTLE, FTLE ( x0 , y0 , Δx,τ ) =

FTLE( x0 , y0 , t )FTLE( x0 + Δx, y0 , t + τ )

σ Aσ B

.

(7)

In the present case, points A and B are aligned along the streamwise position, the relative offset between the two points is Δx=10 mm. Figure 10(a) illustrates the variation of RFTLE, FTLE(x0, y0, Δx, τ) as a function of τ, where point A is located at x0=1050 mm, y0+=50 and point B at xB=x0+Δx=1060 mm, yB+=y0+=50. As shown in Figure 10(a), RFTLE, FTLE reaches its peak at τ=τc, indicating that once positively shifted along time dimension with a step of τc, the time series of FTLE at point B resembles that at point A with the maximum similarity. In this sense, τc may be regarded as the ensemble-averaged time delay 256

that typical LCS takes to convect from point A to point B. If the relative offset Δx between point A and B is small enough, the streamwise component of the averaged convection velocity of typical LCS may be approximated by Δx/τc, namely, Uc≈Δx/τc. Figure 10(b) presents the wall-normal profile of Uc calculated by this method, in which Uc is non-dimensionalized by U∞. The profile of Uc( y) is close to the local mean streamwise velocity profile U( y), especially beyond y+>30. Adrain et al.[23] observed that the convection velocity of the hairpin head is equivalent to the local mean velocity in the outer region of the turbulent boundary layer. The present observation extends their conclusion to the logarithmic layer, further evidencing that the downstream convection of hairpins plays an important role in turbulence transport properties, thus contributing to the mean

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velocity profile of the turbulent boundary layer in the log-region and beyond.

6 Conclusions The present investigation applies the FTLE method to identifying Lagrangian coherent structures in a fully developed flat-plate turbulent boundary layer from a two-dimensional velocity field obtained by time-resolved 2D PIV measurement. Typical LCSs in (x, y) plane revealed by ridges of FTLE field show an inclined streaky pattern with inclination angle θ to the wall, representing the side-view projection of hairpin-like vortices which are characterized as legs of quasi-streamwise vortices extending deep into the near wall region and heads of the transverse vortex tube located in the outer region. The FTLE method may directly reveal the characteristic shape of LCS without any manually selected threshold, and thus is superior to the velocity vector method in objectivity. Statistical analysis on the characteristic shape of instant LCS reveals that the mean value of the inclination angle θ first increases, and then decreases along the 1

Kline S J, Reynolds W C, Schraub F A, et al. The structure of turbu-

wall-normal direction. The probability density distribution of θ accords well with t-distribution in the near wall region, but presents a bimodal distribution with two peaks in the outer region, respectively corresponding to hairpin head and hairpin neck. Spatial correlation calculation of FTLE field is implemented to get the ensemble-averaged inclination angle θR of typical LCS, whose wall-normal variation is similar to that of the mean value of θ. Moreover, the most probable value of θ saturates at y+=100 with the maximum value of about 24°, suggesting that the most likely position where hairpins transit from neck to head is located around y+=100. The ensemble-averaged convection velocity Uc of typical LCS is finally obtained from temporal-spatial analysis of FTLE field. The wall-normal profile of the convection velocity Uc( y) accords well with the local mean velocity profile U( y), evidencing that the downstream convection of hairpins determines the transportation properties of the turbulent boundary layer. The authors would like to express their thanks to the people helping with this work, and acknowledge the valuable suggestions from the peer reviewers. 12

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