Abstract

Introduction
Large-scale coherent structures present in zero pressure gradient (ZPG) turbulent boundary layers have been studied extensively in the past decades and many statistical and struc- tural properties of the flow are well known (Wallace, 2012; Adrian et al. , 2000; Ganapathisubramani et al. , 2005; Hutchins & Marusic, 2007 a ). A remarkable property of the superstruc- tures is their streamwise length, which is on average about 6δ – 8δ. However, instantaneously they can extend up to 10δ – 20δ in the streamwise direction. In addition, they strongly me- ander in the spanwise direction (Eich et al. , 2020; Kevin et al. , 2019) and it has been shown that they can carry a relatively large a portion of the layer's turbulent kinetic energy, espe- cially at large Reynolds numbers (Hutchins et al. , 2011).  In effect, these large-scale structures are the main contribution to the formation of the second plateau/peak in the streamwise ve- locity fluctuations in the log-law layer, which appears at high Reynolds numbers (Fernholz & Finley, 1996; Samie et al. , 2018).  Furthermore, an interaction between superstructures and the near-wall dynamics has been demonstrated (Hutchins & Marusic, 2007b ; Bross et al. , 2019). Therefore, the investi- gation of these superstructures is important for understanding the overall dynamics of turbulent boundary layers. Moreover, it has been shown that turbulent superstructures can modu- late the instantaneous turbulent flow separation line (Eich & Kähler, 2020). In comparison to the large amount of investigations about the coherence of superstructures in the Eulerian reference frame, a Lagrangian description of these structures is largely absent from the literature.  Which is largely due to the technical challenges involved in large-scale Lagrangian measurements of high-Reynolds number turbulent flows.  However, analysis of Lagrangian flow statistics and related coherent structures are important for understanding mixing and mass transport in turbulent systems (Biferale et al. , 2008; Haller, 2015).  This understanding is particularly relevant for engineering applica- tions related to pollutant dispersion, the formation of large- scale meteorological systems, and both industrial and natural mixing processes (Yeung, 2002; Toschi & Bodenschatz, 2009). Therefore, using an experiment involving 12 cameras and 3 overlapping sub-volumes (Schanz et al. , 2019) and the evalua- tion of the data using novel Lagrangian particle tracking meth- ods (Schanz et al. , 2016), a unique data set has been created for the investigation of both Eulerian and Lagrangian descriptions of coherent flow structures.  In the following sections, statis- tics of Lagrangian trajectories within in a turbulent boundary layer subjected to ZPG, FPG, and APG will be analyzed and connected to the presence of elongated superstructures within the flow.
Experimental Setup
The experiments were conducted in the Atmospheric Wind Tunnel Munich (AWM), which is an Eiffel type wind tunnel located at the Universität der Bundeswehr München. The test section is 22 m long and the cross-section area mea- sures 1.85 m × 1.85 m. To achieve a pressure gradient distri- bution, a turbulent boundary layer model, consisting of two S-shaped flow deflections, was installed in the wind tunnel side wall.  The model was designed by Knopp et al. (2018) by means of RANS simulations. In between the flow deflec- tions, a 4 m long flat plate is installed over which zero pressure gradient conditions are present, see figure 1. The flow velocity in this investigation was set to 14 m/s in the bulk flow over the ZPG section, resulting in a Reynolds number, Re τ = 5460 or 1 Re θ = 13 300 in the ZPG region.  Upstream historical effects have been shown to influence the shape/topology of the wake region of this boundary layer but for this Reynolds number it was shown that the turbulent behavior in the near-wall and log- law layer remains largely unaffected (Knopp et al. , 2022) Figure 1: 3D-time-resolved Lagrangian particle tracking experimental setup.  Measurements were preformed in the Atmospheric Wind Tunnel Munich (AWM). The flow was measured using a novel multi-camera multi- volume 3D Lagrangian particle tracking (LPT) technique that was able to capture a flow volume 2.9 m by 0.8 m by 0.25 m in the streamwise ( x ), spanwise ( y ) and wall-normal ( z ) directions respectively.  An imaging system consisting of 12 high-speed cameras was used to record particle images from helium filled soap bubbles (HFSB) tracers illuminated by 10 high-powered LED arrays. This allows for the observation of turbulent super- structures in the ZPG region as they travel through the FPG and APG regions. A sketch of the camera arrangement and field- of-view is shown in figure 1. An exemplary instantaneous plot of the particle trajectories colored with streamwise velocity u is shown in figure 2 which shows the deceleration of the tracer particles in the diffuser region.  The Reynolds stress profiles are provided in figure 3 to illustrate that the near-wall (down to z + ≈ 15) flow properties can be resolved with the LPT mea- surement employed here.  A table of calculated and estimated near and far field flow parameters are given in table 1. The values presented represent an average over the ZPG region. For each run, approximately 2700 time steps were recorded at 2000 Hz. The large effort of planning and perform- ing the measurements were a part of a joint campaign between the UniBw and the DLR. The 3D reconstruction of individual particle tracers from the images recorded by the multi-camera system has been carried out by the DLR using an in-house im- plementation of the Shake-the-Box (STB) algorithm (Schanz et al. , 2016). The post-processing of the trajectory data and further evaluation presented herein was performed by UniBw. More details about the experimental setup, calibration, and La- grangian particle tracking can be found in Schanz et al. (2019). Reynolds stresses in viscous scaling for the ZPG region.  The present experiment (LPT 14 m/s) is at Re τ = 5460. In addition, hotwire (NSTAP) measure- ments at Re τ = 6000 (Samie et al. , 2018) and DNS com- putations at Re τ ≈ 2000 (Sillero et al. , 2013) of a ZPG

Single Particle Lagrangian Dispersion
The dispersion of single particles along trajectories in and around superstructures in a subsection of the entire flow vol- ume is analyzed in this section.  An approximately 1 meter section in the ZPG region was selected to study the trajectories passing through this volume over 150 times steps, correspond- ing to τUε / δ99(ZPG) = 6. The dispersion of a single particle from an initial streamwise release position χ 0 and time t 0 is defined as Δ X i = X i (x 0 , τ) − X i (x 0 , t 0 ) .  An exemplary plot of the dispersion value of subset of trajectories as a function of convective time is shown in figure 4. In figure 5, trajectories in the log-law layer that travel through this region over 150 times steps, are plotted and col- ored with the streamwise fluctuation velocity ι ′ . When a parti- cle traveling along a trajectory exceeds a spanwise dispersion (ΔΥ) value of more that 0.25δ99( ZPG ) , they are color coded in pink.  Since the average Eulerian spanwise spacing of these superstructures is around 0.25δ99(ZPG) (Bross et al. , 2021; No- vara et al. , 2021) and the fact that some of the trajectories dis- perse from their original position more than this value, i.e. the pink color coded locations in figure 5, it can be interpreted that there are mass and momentum exchanges between the super- structures but for the most part single particles remain confined within the superstructures. To quantify the single particle dispersion in the wall nor- mal ( z ) and spanwise ( y ) directions an ensemble (over all data sets) joint-histogram of dispersion in both directions as a func- tion of convective time is shown in figures 6a – 6f. For con- sistency, the maximum count is used to normalize all the other bin values so that the color contour, P goes from 0 to 1.  In addition, the development of the dispersion distribution in the ZPG region (−3000 m ≤ x ≤ −2000 m) is shown in figures 6a − 6c, and the subsequent FPG to APG (−2000 m ≤ x ≤ − 1000 m) is shown in figures 6d − 6f. As the dispersion distribution grows over time it is evident that many trajectories remain close to their initial position at τ = 0. In figure 6a the area inside white contour line (correspond- ing to P = 0.75), is located between |ΔΥ | /δ99( ZPG ) < 0.1 and |ΔΖ|/δ99(ZPG) < 0.05. Therefore, it can be stated that the most probable spanwise and wall-normal dispersion values, over the shown time series, are not larger than the average spanwise su- perstructure spacing of ΔΥ / δ99( ZPG ) = 0.25 (Bross et al., 2019). It could be then interpreted that most trajectories remain rela- tively close (i.e. low dispersion values) to their initial position due to their location within streamwise extended superstruc- tures.  Trajectories with larger dispersion values would then represent trajectories that move in between superstructures. For the FPG/APG region shown in figures 6d — 6f the dispersion distribution in the spanwise direction looks very Figure 6: Joint-histogram of single particle dispersion in spanwise (y) and wall-normal ( z ) directions as a function of convective time for the (a − c) ZPG region (−3000 m ≤ x ≤ −2000 m) and (d − f) and the subsequent FPG to APG region (−2000 m ≤ x ≤ −1000 m). The maximum count is used to normalize all the other bin values so that 0.75. similar to the plots shown for the ZPG region in figures 6a – 6c. This indicates, that in terms of single particle dispersion, the trajectories are not modified in the spanwise direction due to pressure gradients when compared to the ZPG case. This is understandable when the history effects of the proceeding ZPG region are considered (Bobke et al. , 2017) and due to the fact that the superstructures retain their structural features, e.g.  relative spacing, when subjected to strong pressure gra- dients (Eich & Kähler, 2020).  However, in the wall-normal direction, the dispersion distribution is skewed towards posi- tive ΔZ values for the FPG/APG region.  This is likely due to the decelerating flow and thickening of the boundary layer in the APG region as the separation region is approached. In figures 6a – 6f, all trajectories are used to calculated the dispersion distribution, but as the turbulence intensity de- pends on the wall normal location in a boundary layer (see fig- ure 3) the following section will consider dispersion statistics as a function of initial wall normal location.
Structure Function
In order to analyze the statistical behavior of single particle dispersion, the second order structure given by (ΔX;(zo, τ)2) = {(X;(zo, τ) − X;(zo, to))2), as a function of the initial wall normal release location z 0 was computed from an ensemble of all data sets over 150 time steps for ΔΥ and ΔΖ. Since the turbulent fluctuation intensity depends on the wall normal position, see figure 3, the second order structure func- tions were computed as a function of the initial wall normal release position, z 0 , in 10 mm bins (0.06/ δ 99 ( Z PG ) and plot- ted in figures 7a and 7b. In these plots the reference location τU ∞ / δ99(ZPG) = 0 corresponds to x = −2000 m, i.e. negative τ corresponds to the dispersion in the ZPG regions and positive τ corresponds to the FPG and APG region. Figure 7: Second order structure function of single parti- cle dispersion in the (a) spanwise and (b) wall normal di- rections vs. τU ∞ / δ99( ZPG ) for different initial release lo- cations in the wall normal direction z . Wall normal bins (zo) have width of 10 mm or 0.06/ δ99(ZPG) ·  Reference location τU∞ / δ99( ZPG ) = 0 corresponds to x = −2000 m so negative τ corresponds to the dispersion in the ZPG regions and positive τ corresponds to the FPG and APG region. In figure 7a values of the second order structure func- tion, 〈ΔΥ〉 2 , at z 0 bins near the wall increase until z 0 ＝ 0.18899(ZPG) - 0.23899(ZPG) and then gradually decrease as z 0 bin values approach the wake. This is also true for 〈ΔΥ〉 ² in the FPG/APG region shown in figure 7a.  Since these bin are located in the log-law region, where the highly energetic super- structures are centered in, the maximum 〈ΔΥ〉 ² values coming from trajectories released in the log-law layer is understand- able. The spanwise meandering of the superstructures is likely responsible for the elevated {ΔΥ} 2 values.  Furthermore, the spanwise structure function, {ΔΥ} 2 , looks almost symmetric about τ = 0. In other words, its appears that spanwise disper- sion is not modified from the ZPG to the FPG/APG regions. This is likely due to the fact that the Eulerian spanwise super- structure topological properties are also preserved in the strong APG flow following the ZPG region (Eich & Kähler, 2020; Bross et al., 2021). As previously indicated by the joint-histogram analysis if figure 6f, the wall-normal structure function ( {∆Z} ² ) is starkly different in the FPG/APG (τ > 0) than the ZPG region. Inter- estingly, at around τUε / δ99( ZPG ) ＝ 4, bins near the wake region overtake the {ΔΖ} 2 value of bins closer to the wall.  Which means that, on average, trajectories originating in the wake re- gion initially have less wall normal dispersion than trajectories closer to the wall, but after several eddy turn over times, those trajectories disperse more significantly in the wall normal di- rection than ones closer to the wall. In figures 8a and 8b, {ΔΥ} 2 in the ZPG region is plotted in inner/viscous scaling while the time lags are scaled with in- ner/viscous and boundary layer or eddy turnover time T L . In both plots, the axis are shown with a log-log scaling in or- der to better see short time lags ( τ < T L ). For short time lags ( τ < T L ) the second order structure function appears to vary quadratically with time and for larger time lags (τ > T L ) the relationship appears linear.  This is consistent with the pre- dictions of Taylor (Taylor, 1922) for homogeneous isotropic turbulence (HIT).
Conclusions and Outlook
In this work, the 3D-dimensional flow structure of an evolving turbulent boundary layer from ZPG to FPG to APG and eventually flow separation, was studied using a novel large-FOV Lagrangian particle tracking technique. The large- volume allowed for the simultaneous tracking of large-scale turbulent superstructures in the streamwise, spanwise, and wall-normal directions.  In addition to the average structure properties, this data set can be used to investigated the tempo- ral evolution of superstructures. Furthermore, the availability of Lagrangian trajectories allows for the investigation of how individual trajectories behave and their relationship with the turbulent superstructures. It was found that the dispersion of single particles along trajectories in the log-law layer are capable of motions that ex- tend past the average Eulerian superstructure spacing in the spanwise direction.  Which suggests that there is mass and momentum exchange between the high and low momentum superstructures.  Moreover, it was found that the most proba- ble dispersion values after roughly 6 boundary layer turnover times lie within an area corresponding to the average Eulerian superstructure width. The same conclusion can be made for the spanwise dispersion values in the FPG/APG region, sug- gesting that the superstructure topology is not modified from ZPG to APG for this model geometry. Looking at the mean square or second order structure function of single particle dispersion, it is clear that maximum spanwise dispersion values occur within the log-law layer for both ZPG and FPG/APG regions.  Furthermore, the scaling predicted by Taylor for HIT also appears to remain valid for the present case despite the highly aniostropic nature of a tur- bulent boundary layer. In the future, analysis of the local trajectory path is needed in order to better identify full or parts of trajectories that lie in- side or between coherent structures.  Analysis of integral path lines, individual trajectory correlations, and dual particle dis- persion might provide further information for the detection of Lagrangian coherent structures based on the characteristics of individual and groups of particle trajectories within the flow.
Acknowledgments
This work is supported by the Priority Programme SPP 1881 Turbulent Superstructures funded by the Deutsche Forschungsgemeinschaft project numbers KA1808/21 and SCHR1165/5. In addition, this work was partially funded by the DFG project number KA1808/14.