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1 Introduction

Jiménez [5] showed that eddies with streamwise lengths of 10–20 \(\updelta \) are present in the logarithmic region of wall-bounded flows by compiling results from existing measurements and numerical simulations. Kim and Adrian [2] found streamwise energetic modes with wavelengths up to 14 pipe radii within fully developed turbulent pipe flow. They accounted the alignment of packets of hairpin-vortices as responsible for the creation of such structures and termed them ‘very large scale motion’ (VLSM).

Indications of the existence of similar flow structures within the log-region of turbulent boundary layers (TBLs) were given by Tomkins and Adrian [6], as well as Ganapathisubramani et al. [7]. They were able to document the existence of long stripes of negative or positive streamwise velocity fluctuations (u’) within these domains. Both publications relied on measurements using the method of particle imaging velocimetry (PIV) with a streamwise length of the investigation area around two times the boundary layer thickness (\(\updelta \)). Therefore the full extent of the found structures could not be examined.

Hutchins and Marusic [1] and again Marusic et al. [8] confirmed these results using a similar PIV-setup. Additionally they performed measurements using a hot-wire rake, covering a spanwise distance of more than one \(\updelta \) with eleven hot-wire probes. By applying Taylor’s hypothesis on the obtained time-series, they were able to extract quasi-instantaneous snapshots of the flow structures at several heights above the wall. In many of these snapshots very long structures of positive and negative u’ can be seen, frequently exceeding a length of 20 \(\updelta \). These regions of negative and positive u’ typically appear besides each other and show a meandering behavior in spanwise direction. The authors account this meandering for the fact that the length scales indicated by single-point statistics were much shorter (around 6 \(\updelta \)). Due to the large extent of the found features, the authors termed them ‘superstructures’.

As shown by DNS data [4], these superstructures seem to directly interact with small-scale structures near the wall (leaving a ‘footprint’). This impact on the conditions near the wall is of particular interest, as the large-scale structures underlie outer scaling (their extent is dependent on \(\updelta \) and therefore also the Reynolds number), while it was assumed that near-wall structures do not. The interaction of large and small-scale structures challenges this assumption. A further examination of the formation and evolution of superstructures in different flow conditions is therefore of interest for a better understanding of turbulent flows in general.

Rahgozar and Maciel [3, 4] investigated large and very large-scale structures in a turbulent boundary layer subjected to a strong adverse pressure gradient (APG). Horizontal measurement planes with a height of 3 \(\updelta \) at different heights and streamwise locations were investigated using PIV. It was found that the general features of large flow structures are retained under the presence of an APG, however the frequency of appearance decreases. Especially in the lower parts of the boundary layer the appearance of high- and low-speed streaks diminishes in comparison to a zero-pressure-gradient case.

For this study, large-scale PIV-data of a TBL within a slowly rising adverse pressure gradient is analyzed with respect to the sizes of the occurring structures; indications of the presence of superstructures within this flow regime are gathered.

2 Experimental Setup

The experimental data was gained in context of the DLR-internal project ‘Rettina’. The aim of the project was to generate an experimental dataset suitable for the extraction of possible enhancements and/or modifications of the law of the wall for flows within a pressure gradient [9, 10].

A large model was designed that allowed maximizing the achievable Reynolds number, smooth changes of the pressure gradient and good measurability of the flow properties. Figure 1 shows the model in top view. Spanning over 12 m, it consists of a long flat plate with superelliptic nose that allows the buildup of a TBL of significant thickness. The deflection is achieved by a 1.5 m-long arc with large curvature radius, followed by a 0.8 m long flat plate, inclined at an angle of 13\(^\circ \) relative to the main flat plate. The flow is directed back by another arc, followed by a flat plate. At the end of the model a 1.5 m-long flap can be used to ensure symmetric flow conditions at the nose.

An appropriate wind tunnel was found with the ‘Atmospheric Wind Tunnel’ at the Universität der Bundeswehr in Munich. The 22 m long test section with a cross section of \(2 \times 2\,\mathrm{{m}}\) allowed for huge dimensions of the model, wind speeds up to 40 m/s are achievable; the measured turbulence level is 0.1 %.

In order to be able to describe the development of the flow characteristics for all pressure regions—from the undisturbed boundary layer, over the onset of the deflection to the fully developed pressure gradient—a large scale PIV-setup was used.

Eight high-resolution PCO 4,000-Cameras with 11 Megapixel each were installed in series on top of the wind tunnel, observing a total region of over 3 m in the vicinity of the APG. Figure 2 shows the approximated fields of view of all cameras. Seven cameras used 100 mm-lenses, while one camera, connecting the main areas of interest, used a 50 mm-lens. The described camera system allowed retaining a high spatial resolution, in spite of the very large investigation area. For the work presented in this chapter, only the four cameras downstream (labeled 1–4) were used.

Fig. 1
figure 1

Top view of the model installed upright in the center of the wind tunnel, with flow being directed past both sides. The flap at the end is used to compensate for circulation around the model

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Fig. 2
figure 2

Fields of view for the eight PCO 4,000 cameras. The four cameras used within this work are numbered 1, 2, 3, 4 in streamwise direction

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Illumination was realized using two partially overlapping light sheets, generated by two double pulse lasers. The flow was seeded with DEHS droplets, having an average diameter of approx. \(1 \upmu \mathrm{m}\). The gained images were correlated using an iterative multigrid algorithm with image deformation. A final window size of \(24 \times 24\) pixels (\(16 \times 16\) in case of camera 1), combined with an overlap of 66 %, yields a vector spacing of approx. 0.65 mm (1 mm for camera 1). A more detailed description of the experiment can be found in [11].

Fig. 3
figure 3

Instantaneous snapshot of u-velocity for U \(=\) 6 m/s (stitch of four camera views)

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The calculated vector fields for each camera are combined according to the calibration to yield one continuous vector field of approx. 1.7 m length and around 20 cm wall-normal height. Figure 3 shows an example of the instantaneous velocity field as seen by the four cameras considered.

Fig. 4
figure 4

Averaged u-velocity for a free stream velocity of \(\mathrm{U} = 6\,\mathrm{m}/\mathrm{s}\). Streamtraces starting at \(\mathrm{y} = 0.04 \updelta (\mathrm{I}), \mathrm{y} = 0.17 \updelta (\mathrm{II})\) and \(\mathrm{y} = 0.32 \updelta (\mathrm{III})\)

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Using a free stream velocity of \(\mathrm{U} = 6\,\mathrm{m}/\mathrm{s}\), a boundary layer thickness of \(\updelta _{99} = 106\,\mathrm{mm}\) was measured at the end of the long flat plate. Around 6,000 independent snapshots were taken at a frequency of 1 Hz, allowing for converged statistics. Figure 4 shows the averaged u-velocity.

3 Structure Characterization

Looking at instantaneous streamwise fluctuations (see Fig. 5) it is obvious that patterns resembling the sought-after superstructures can be identified. In the rear part of the APG a stretched region of slow fluid can be seen that spans a length of around 80 cm (\({\sim }8 \updelta \)), before it exists the investigation area. Evidence for similar structures can be found in many instantaneous snapshots, though the features often seem to be more disturbed and broadened compared to the hot-wire results in [1]. As can be seen in Fig. 5, the structures are mostly angled to the (tilted) wall and seem to be stretched in the direction of the main flow. This inclination does not seem to change noticeably with progression of the pressure gradient.

Fig. 5
figure 5

Instantaneous streamwise velocity fluctuations, derived from the velocity field shown in Fig. 3

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As to be able to describe the structures’ evolution within the growing influence of the APG, streamtraces were extracted from the averaged velocity data. Along these streamtraces, two-point correlations of streamwise velocity fluctuations (\(\mathrm{R}_\mathrm{uu}\)) were performed in order to extract general features of the occurring structures and their development in time. The fixed points of the two-point correlations were positioned at 45 equidistant points following along the streamtraces. This approach was chosen, assuming that occurring structures should mostly follow the mean flow. Three streamtraces with starting points at different heights above the wall (\(\mathrm{y} = 0.04 \updelta \;(\mathrm{I}), \mathrm{y} = 0.17 \updelta \;(\mathrm{II})\) and \(\mathrm{y} = 0.32 \updelta \;(\mathrm{III})\)) were examined—see Fig. 4.

Fig. 6
figure 6

Two-point correlation of streamwise velocity fluctuation (\(\mathrm{R}_{\mathrm{uu}}\)) for a fixed point on streamline II in the early stages of the adverse pressure gradient

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Fig. 7
figure 7

Two-point correlation of streamwise velocity fluctuation (\(\mathrm{R}_{\mathrm{uu}}\)) for a fixed point on streamline II in the later stages of the adverse pressure gradient

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Figures 6 and 7 show exemplary correlation results for two fixed points on streamtrace II. The correlation plane at the onset of the APG (Fig. 6) shows a streamwise elongation of around 250 mm, indicating the presence of structures with an average length of 2–3 \(\updelta \). The inclination of the correlation figure is around \(10^{\circ }\) relative to the main flow direction. In case of an undisturbed TBL, the value of this inclination would be 11–13\(^{\circ }\).

The characteristic shape of the correlation figure stems from the presence of hairpin-like vortices, which are typically inclined at an angle of around \(45^{\circ }\) relative to the wall (see e.g. [12]). The averaging of the occurring structures, as done by the process of two-point-correlation, results in the elliptical shape with inclination towards the wall (see [1] for an example in a zero-pressure-gradient case).

Looking at the results at a later stage of the APG (Fig. 7), it is evident that the correlation is broadened and the peak is smeared out. The occurring structures are stretched in length by the force of the APG and widened by the increasing turbulence within this region. Still, the inclination of the correlation figure is fixed at a value of approx \(10^{\circ }\). The structures may be convected with the flow, as well as deformed by the occurring forces, but their general direction and orientation remains unchanged.

In order to quantify the development of structure sizes within the APG, the integral length scale of all considered two-point-correlations was calculated:

$$\begin{aligned} L_{{{{uu}}}} \left( r\right) = \int \limits _{{- \infty }}^{\infty } {R_{{{{uu}}}} \left( r\right) ds} \end{aligned}$$

This value gives a measure of the averaged elongation of the structures along the direction of the line integral. In this case, the line integral was chosen to run along the extracted streamtrace (as indicated in Figs. 6 and 7). \(\mathrm{L}_\mathrm{uu}\) was evaluated for all 45 fixed points of the two-point correlations along streamtraces I, II and III, giving the development of structure sizes within the growing influence of the APG.

Figure 8 shows the results of these calculations. In case of streamtrace I, which is positioned close to the wall (\(\mathrm{y} = 0.04 \updelta \)) at the upstream edge of the interrogation volume, the structures clearly gain in size while following the streamtrace: averaged structure sizes of around \(1.6\,\updelta \) are found at the beginning (at \(x \approx \) 7300 mm), which increase to around 2.6 \(\updelta \) at the end of the interrogation volume (at \(x \approx \) 8700 mm). The streamtrace is clearly lifted from the model wall by the APG, the flow being on the verge of detachment. The smaller-scale structures near the wall are stretched significantly during this process and gain in size.

Fig. 8
figure 8

Spatial development of the integral length scale for two-point-correlations along streamtraces I (left), II (middle) and III (right)

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This effect is less pronounced in the regions higher up the TBL: the increase in structure size is far less pronounced in case of the other streamtraces. For streamtrace II initial structure sizes of around 2.2 \(\updelta \) can be seen, which rise to a maximum of around 2.8 \(\updelta \) at the end of the interrogation volume. For streamtrace III the initial size of approx. 2.8 \(\updelta \) rises to around 3.2 \(\updelta \) in the final stages of the APG. The increase in structure size seems to be connected to the relative growth of wall distance with the course of the streamtrace. Actual structure sizes may be slightly underestimated by this approach, as the maximum extent of the correlation figure is not along the streamtrace, but along the inclination relative to the main flow. It has to be stressed that very long structures are singular events that cannot be adequately captured by two-point correlations, as these constitute averages of all occurring flow phenomena. Additionally, large structures are often not imaged entirely due to spanwise meandering, reducing the perceived structure size.

4 Conclusion and Outlook

By performing successive two-point-correlations along streamtraces of the flow under the influence of an adverse pressure gradient, it was possible to document the development of averaged structure-sizes and-orientation. It can be seen that structures spanning multiple \(\updelta \) can be found on a regular basis, which grow in size with increased distance to the model wall. The presence of the pressure gradient both stretches and widens the flow features.

With respect to the sought-after super-structures these investigations showed to be not suitable to extract a great deal of relevant information. Though it is clear that very long structures persist within the APG-region (see Fig. 5), their occurrence is irregular; statistical approaches, such as two-point-correlations, are biased by smaller structures. Therefore, the real length of such structures (\(>\)8 \(\updelta \) by visual inspection of the data, 20 \(\updelta \) as given in [1]) could not be confirmed in the scope of the conducted investigations.

One obstacle in the observation of super-structures in the available dataset is the spanwise meandering of the structures, described in [1]. As the PIV measurement plane is oriented in wall normal direction it is possible that existing super-structures move in and out of the measurement plane within one single snapshot. If this is the case, the real extent of the structures will be masked to the measurement. However, for a subset of the measurements the meandering is of such a small scale that the whole structure is visible. In order to identify such cases and to extract viable information, other examination methods are needed, such as conditional averaging or pattern-recognition, similar to the methods applied in [3].

In the scope of such an investigation, the additional data from the four upstream cameras, observing the undisturbed boundary layer and the early onset of the APG, would be included. By this, occurring structures could be captured at their full length, and the whole process of structure development within the pressure gradient could be described.

Furthermore, measurements on the same model using tomographic PIV [13] are planned. These investigations could simultaneously show the meandering tendency of the occurring structures, as well as document their wall-normal appearance.