2.2.1. Materials and PIV equipment

A basic premise in PIV is that tracing particles faithfully follow the paths of the water elements they displace. We seeded our flow with hollow glass particles. Their shape was nearly spherical, their mean diameter was $d_{p}=35~{\rm\mu}\text{m}$ , and their density had a nominal value of ${\it\rho}_{p}=1080~\text{kg}~\text{m}^{-3}$ . Whereas using particles heavier than water generally results in underestimation of the velocity magnitude, the particle properties used in this study were a compromise, selected to satisfy a variety of experimental constraints. The ratio of the particle response frequency ( $t_{p}^{-1}$ ) to that of the characteristic flow frequency is known as the particle Stokes number: $\mathit{St}=({\it\rho}_{p}{d_{p}}^{2}/18{\it\mu}t_{f})$ , where ${\it\mu}$ is the dynamic viscosity of water and $t_{f}$ is a time scale for the turbulent velocity fluctuations in the flow. Calculated $\mathit{St}$ were 0.0086, 0.0098 and 0.0122 for the LRe, MRe and HRe cases, respectively, based on $t_{f}$ values of $9.2\times 10^{-3}$ , $8.1\times 10^{-3}$ and $6.5\times 10^{-3}~\text{s}$ . These values of $t_{f}$ are the characteristic turn-over times of the smallest, distinct and clearly identifiable vortices for each $\mathit{Re}$ case. They are considered representative of the time scales of flow fluctuations that are of particular interest in this study. On the other hand, the particle time scale had a value of $t_{p}=8.18\times 10^{-5}~\text{s}$ . It is, therefore, expected that seeding particles would respond with negligible lag to changes in the direction of the bulk flow.

A 20 W dual-cavity pulsed Nd:YLF laser was used to illuminate the flow. Each pulse had a wavelength of 527 nm, was fired at a frequency of 1000 Hz, and emitted a nominal energy of 10 mJ. Using a convex lens with a focal length of 50 mm, a spherical lens with a focal length of 300 mm, and a series of mirrors, the beam was delivered in the test section through the transparent bottom of the channel (figure 1). The thickness of the laser sheet was $2.0\pm 0.2~\text{mm}$ throughout the camera’s field of view. We selected this thickness to effectively deal with out-of-plane particle motion: an important source of error in PIV measurements.

To achieve the desirable magnification of the region of interest, the distance between the centre of the camera lens and the CMOS sensor was adjusted using extension tubes. The optical axis of the lens was perpendicular to the channel’s sidewall to eliminate the unwanted effects stemming from refraction of light due to its propagation through different media (air, Plexiglas, water). The final geometric configuration resulted in a maximum error of 1.35 % affecting the outermost regions of the images due to deviations of the viewing angle from the optimum of $90^{\circ }$ . Autocorrelations of image intensities yielded an average diameter for the imaged particles of $d_{{\it\tau}}=2.3\pm 0.5~\text{pixel}$ . A value close or higher than two pixels has been reported as sufficient to drastically mitigate pixel-locking-induced errors (Raffel et al. Reference Raffel, Willert, Wereley and Kompenhans2007; Adrian & Westerweel Reference Adrian and Westerweel2011).

The resulting depth of field was $3.2\pm 0.2~\text{mm}$ ( ${>}$ light sheet thickness, so that diffraction-limited images are captured) and the average magnification factor was equal to $63.52\pm 0.19~{\rm\mu}\text{m}/\text{pixel}$ . Particle motion was tracked with a $1.3\times 10^{6}~\text{pixel}$ high-speed camera, which was operated at a frequency of 1000 Hz. The inter-frame time of 1 ms assured maximum particle displacements between four and seven pixels for all $\mathit{Re}$ cases. We selected the single-frame mode for PIV data acquisition to: (a) obtain maximum light intensity by the simultaneous firing of the two laser cavities, (b) maximize the number of collected images, and (c) minimize loss of correlation due to unequal exposures between consecutive images. As a result, 3270 frames filled the 4 GB-buffer of the camera for each $\mathit{Re}_{D}$ case. The total sampling time of 3.27 s corresponded to an average of $24.5\pm 1.4$ characteristic times ( $t_{c}={\it\delta}_{99}/U_{0}$ ) of the incoming flows. The maximum available image resolution of $1280~\text{pixel}\times 1024~\text{pixel}$ was used. Consequently, the average dimensions of the imaged flow field was the same for all three experiments, $7.31~\text{cm}\times 5.62~\text{cm}$ .

2.2.2. Image preconditioning and analysis

Image preconditioning included the subtraction of the time-averaged mean image intensity from every frame (Honkanen & Nobach Reference Honkanen and Nobach2005). Additional enhancements were achieved by accurately defining the boundaries of the domain through gradient-based edge detection. To account for the undesirable contributions of some very bright particles in the resolution of the domain, we capped their intensity (Shavit, Lowe & Steinbuck Reference Shavit, Lowe and Steinbuck2007).

We adopted a common analysis procedure for all three $\mathit{Re}_{D}$ cases. PIV images were processed with software that implements the Robust Phase Correlation algorithm (Eckstein, Charonko & Vlachos Reference Eckstein, Charonko and Vlachos2008; Eckstein & Vlachos Reference Eckstein and Vlachos2009). Multi-grid, iterative and deformable windows were used to resolve the instantaneous flow fields. Three passes with two iterations in each were applied. The dimensions of the interrogation windows were reduced in each subsequent pass (from $64~\text{pixel}\times 64~\text{pixel}$ to $32~\text{pixel}\times 32~\text{pixel}$ to $16~\text{pixel}\times 16~\text{pixel}$ ). Each window overlapped by 50 % with its adjacent ones in both $x$ and $y$ directions. Discrete Window Offsetting (Westerweel Reference Westerweel1997) accounted for the loss of correlation signal in the second pass due to in-plane particle motion. Vector validation was performed based on velocity thresholding and the Universal Outlier Detection technique (Westerweel & Scarano Reference Westerweel and Scarano2005). The percentage of valid vectors derived from the PIV image analysis ranged from 96.5 % to 99.7 %. The final processed velocity fields had a uniform vector grid spacing of $8~\text{pixel}\times 8~\text{pixel}$ . On average, the Eulerian representation of the instantaneous, two-dimensional flow field was obtained using 17 828 velocity vectors. The total number of measurements collected for all three $\mathit{Re}_{D}$ cases was 174 889 410.

2.2.3. Data post-processing

Velocity derivatives were calculated using a compact fourth-order scheme with Richardson extrapolation. This hybrid operator has been shown to outperform conventional second-order finite difference schemes for the range of wavenumbers typically present in PIV data (Etebari & Vlachos Reference Etebari and Vlachos2005). Note, however, that some discrepancies in PIV vorticity data will always exist regardless of the selection of the differential operator. These stem from contributions of the very small turbulence scales that remain under-resolved (Wallace & Foss Reference Wallace and Foss1995). Since this work focuses on the behaviour of larger turbulence scales, we expect that this effect is negligible.

3.2.1. Case of $\mathit{Re}_{D}=2.9\times 10^{4}$ (LRe)

The time-resolved analysis of flow dynamics for the LRe case revealed numerous flow patterns that were not detectable in the time-averaged velocity signals. High levels of intermittency characterize this junction flow, manifested in both large (organized vortices) and small-scale flow components (near-wall fluid flow). For example, we discovered that the junction region is home to a number of vortices ranging from one to four (figure 8 a,b). Vortex stretching and meandering increase the spatial extent of the orbits followed by the vortex cores. It is possible, therefore, that the secondary vortex (HV2) shown in the topology of figure 2 is just the average expression of a number of vortices, instead of a unique flow structure. More confidence exists in the expression of the primary vortex, although its presence on the flow field is not continuous.

Figure 8. Characteristic instantaneous flow topologies for the LRe case showing: (a) the minimum and (b) the maximum number of vortices.

The occasional disintegration of HV1 is a feature for which no hint was provided for by the time-averaged analysis. It was the frame-by-frame inspection of the flow field for the LRe case that revealed it. For a significant number of frames, the flow topology was devoid of the primary horseshoe vortex. These snapshots were clustered in thirteen groups. Their total duration was 440 ms, representing 13.5 % of the LRe experiment. The disappearance of HV1 is interpreted as a loss of its coherence. Figure 9(a–d) shows that this state is preceded by the collision of HV1 with a fluid patch originating from the downwash flow parallel to the obstacle. This mechanism could offer a plausible explanation for the obliteration of HV1, because it was identified in the majority of the episodes (eight out of thirteen). The absence of this or another possible mechanism (Paik et al. Reference Paik, Escauriaza and Sotiropoulos2007) for the remaining five episodes, however, casts doubts on the generality of that conclusion. Regardless of the origin of the phenomenon, the unsteadiness in the region formerly occupied by the core of HV1 is substantial (figure 9 c). The flow dynamics locally appear to assume an erratic behaviour, with changes in topology occurring even faster than the sampling rate of 1000 Hz. This also prohibits drawing sound conclusions about what brings about the end of the unsteadiness and re-stabilizes the flow, which then returns to more easily identifiable patterns (figure 9 d). Note that during the absence of HV1, the secondary vortex retains its coherence and relative position within the flow topology.

Figure 9. Sequence of snapshots displaying the disintegration of HV1 for the LRe case. Flood contour maps express the magnitude of the wall-normal velocity component. Dimensionless time is indicated by the value of $t^{\ast }$ : (a)  $t^{\ast }=7.43$ , (b)  $t^{\ast }=7.53$ , (c)  $t^{\ast }=7.63$ , (d)  $t^{\ast }=7.73$ .

The discussion on the dynamics of the primary horseshoe vortex is extended to include interactions with HV2. In the absence of the tertiary vortex, there is a direct interplay between HV1 and HV2. Figure 10(a–d) demonstrates a sequence of snapshots describing an episode that ends up with the amalgamation of the vortices. In the beginning (figure 10 a), the two cores are close together. In a subsequent frame (figure 10 b), the two structures share a number of common streamlines at the lower edges of their peripheries. The interaction becomes closer, so that both cores are surrounded by the same outer streamlines (figure 10 c). Observe also the appearance of a new, transient structure in the vicinity of the cylinder. Finally, the two vortices merge into a single, large-scale structure (figure 10 d).

Figure 10. Amalgamation of HV1 with HV2: (a)  $t^{\ast }=7.85$ , (b)  $t^{\ast }=7.87$ , (c)  $t^{\ast }=7.93$ , (d)  $t^{\ast }=8.08$ .

Next, we revisit the bimodal dynamics of the horseshoe vortex motivated by the prominent position of this subject in the junction flow literature (Devenport & Simpson Reference Devenport and Simpson1990; Paik et al. Reference Paik, Escauriaza and Sotiropoulos2007; Kirkil & Constantinescu Reference Kirkil and Constantinescu2009). Instantaneous representations of the global velocity field simplify flow classification. Thus, there is no need to resort to indirect methodologies (like the one devised by Devenport & Simpson (Reference Devenport and Simpson1990) for data collected with point-wise velocity measuring techniques) to determine the characteristic modes and the time spent by the flow in each one of them. A painstaking examination of the cinematography of the PIV data suffices for a direct and unambiguous detection of these modes.

A noteworthy finding of our analysis is that zeroflow and backflow are not the only modes that characterize junction flows. For a significant number of frames, the characteristics of the flow field do not verify the topologies of any of these two well-known modes. Figure 11(a–c) juxtaposes three snapshots representative of the backflow, the zeroflow and this other mode, which we will refer to from now on as intermediate. The backflow mode is characterized by a strong near-wall jet, which moves against the bulk flow (figure 11 c). On the other hand, zeroflow mode occurs when the motion of the jet is impeded (figure 11 a). The reverse flow is then directed upwards at high velocities. These descriptions are in good agreement with the original definitions provided by Devenport & Simpson (Reference Devenport and Simpson1990) and have been corroborated by subsequent studies (Paik et al. Reference Paik, Escauriaza and Sotiropoulos2007; Kirkil & Constantinescu Reference Kirkil and Constantinescu2009; Gand et al. Reference Gand, Deck, Brunet and Sagaut2010). Differences also emerge through this comparison. The most striking of them is that, in contrast to what has been suggested in the literature, the flow modes do not have preferential locations within the flow domain. For example, in figure 11(c) one would have expected the appearance of zeroflow, because the primary horseshoe vortex is located very close to the cylinder. Nevertheless, the backflow mode is clearly present. As far as the intermediate mode is concerned, it collectively symbolizes every flow topology that does not fall into one of the well-known categories. Although there are various expressions of the intermediate mode, figure 11(b) highlights a typical one. It is observed that the interface between the incoming and return flow does not form a vertical line. The rather weak return flow separates from the bed and is directed upwards, but is lacking the momentum required to propagate further upstream. Its path terminates upon its encountering with the incoming flow. Unlike the zeroflow mode, the incoming flow here is not strong enough to cause the vertical ejection of fluid at high velocities. Essentially, the momentum of the return flow is diffused within its surroundings. Previous studies have mentioned the possibility of existence for additional junction flows modes, but did not elaborate further on the subject (Devenport & Simpson Reference Devenport and Simpson1990; Praisner & Smith Reference Praisner and Smith2006a ). The intermediate mode indicates that the switching between the two modes is not continuous. This could potentially suggest modifications in the nature of phenomena that have been linked to the mode-switching mechanism (Agui & Andreopoulos Reference Agui and Andreopoulos1992; Ölçmen & Simpson Reference Ölçmen and Simpson1994; Praisner et al. Reference Praisner, Seal, Takmaz and Smith1997; Kirkil & Constantinescu Reference Kirkil and Constantinescu2009; Escauriaza & Sotiropoulos Reference Escauriaza and Sotiropoulos2011a ,Reference Escauriaza and Sotiropoulos b ).

Figure 11. Characteristic junction flow modes: (a) zeroflow; (b) intermediate; and (c) backflow. Flood contours plot the magnitude of the streamwise velocity component.

The sequence of the three modes appears to be random. Zeroflow mode is present 41.5 % of time. The intermediate mode follows with 33.8 %. Backflow mode occurs only during 11.2 % of the total duration of the LRe case. As mentioned previously, for 13.5 % of the time the disorganization of the primary vortex does not allow classification to any of the above flow modes. The average frequencies of appearance, made dimensionless after being multiplied with $t_{c}$ , are as follows: 0.86 for the zeroflow mode, 1.12 for the intermediate mode and 0.52 for the zeroflow mode. The latter mode is the most short-lived and the one that appears less frequently during the LRe case.

3.2.2. Case of $\mathit{Re}_{D}=4.7\times 10^{4}$ (MRe)

A characteristic snapshot of the instantaneous flow topology for the MRe case is presented in figure 12. A thorough inspection of the cinematography revealed that the maximum number of vortices appearing simultaneously on a snapshot was five and the minimum was one (just the primary vortex). Throughout this experiment, HV1 dominated the flow field. In general, the topologies of the MRe case were well-defined and bore resemblance to topological models representing time-averaged flows. Observe for example the characteristic sequence of vortices in figure 12. The symmetrical configuration comprises two pairs of clockwise/counterclockwise vortices: HV1 with HV3 and HV2 with HV4. Most of these topologies, however, cannot be sustained for a substantial amount of time. Momentum and vorticity fluctuations in the incoming boundary layer or interactions between the vortices drive the flow into more stable states.

Figure 12. Instantaneous flow topology for the MRe case demonstrating the simultaneous presence of four vortical structures. Flow streamlines are superimposed on flood contours of the magnitude of the streamwise velocity component.

We then shift the focus to the study of the downflow that was shown (figures 2 and 3) to be a notable feature of the MRe case. Video animations (figure 13 a–c) revealed that patches of negative vorticity run down the face of the cylinder in a nearly uninterrupted fashion. They are advected on top of the continuous layer of positive vorticity originating very close to the cylinder’s face as a result of fluid–structure interactions. Occasionally, two counter-rotating blobs of vorticity emerge and run side-by-side downwards (figure 13 a). These structures do not have a single origin: some of them had previously travelled with the incoming boundary layer flow, others entered through the upper part of field of view, and, finally, a third category formed right at the cylinder’s face when horizontal flow impinged on the obstacle. Once fluid parcels of negative vorticity approach the bed, they join the return flow directed upstream (figure 13 c). This change of direction is accompanied by the generation of positive vorticity very close to the wall. From this point on, the positive patch of vorticity moves parallel to the negative patch of vorticity and towards the upstream. Eventually, they merge respectively with the elongated patch of positive vorticity below HV1 and the core of HV1 (negative vorticity), strengthening the rotationality of these flow components. This demonstrates that the eruptions of near-wall fluid do not originate exclusively from the interaction between HV1 and the wall. Some of them are simply high-vorticity fluid masses making their way towards HV1 through the jet of return flow.

Figure 13. Sequence of frames illustrating the transport of two patches of opposite spanwise vorticity within the junction: (a)  $t^{\ast }=17.22$ , (b)  $t^{\ast }=17.47$ , (c)  $t^{\ast }=17.80$ .

The dynamics of the tertiary vortex can be more clearly tracked within the cinematography of the MRe case. As shown in figure 14(a), the tertiary vortex originates from two sources: the approach boundary layer and the return flow of fluid that had previously impinged on the cylinder. In agreement with findings from the time-averaged analysis, the relevant positions of HV1 and HV2 also play a role in the formation of HV3. The dynamic interplay of these flow components determines the location of the tertiary vortex in time. In particular, in figure 14(b), HV3 is stretched and acts as a barrier separating the primary vortex and the reverse flow jet from the incoming flow. The tertiary vortex also traps the positive vorticity that is transported with the reverse flow jet (figure 14 c). This modifies the wall-eruption mechanism (Paik et al. Reference Paik, Escauriaza and Sotiropoulos2007), resulting in weaker extractions of near-wall fluid than those observed for the HRe case. Nevertheless, occasionally these events can bring about the disintegration of the tertiary vortex within the MRe experiment (figure 14 d).

Figure 14. Dynamics of tertiary horseshoe vortex (HV3), indicated by the black arrow. The background displays the instantaneous magnitude of out-of-plane vorticity. The figure is annotated with the dimensionless circulation, $|{\it\Gamma}^{\ast }|$ , associated with the individual vortices: (a)  $t^{\ast }=14.29$ , (b)  $t^{\ast }=14.73$ , (c)  $t^{\ast }=15.01$ , (d)  $t^{\ast }=15.44$ .

This subsubsection concludes with information regarding the characteristic flow modes. Time-wise, they are nearly equivalent for the MRe case. In particular, the percentages of time for which the zeroflow, the intermediate and the backflow modes appear on the instantaneous flow snapshots are respectively: 31.7 %, 36.1 % and 32.3 %. The dimensionless frequencies of appearance reveal a slightly different picture and measure: 0.91 for the zeroflow, 0.96 for the intermediate and 0.72 for the backflow mode. Therefore, for the MRe case, the intermediate mode is the one appearing more frequently and for a longer duration compared to the other two modes.

3.2.3. Case of $\mathit{Re}_{D}=12.3\times 10^{4}$ (HRe)

Video animations revealed qualitative similarities between the instantaneous topologies of the HRe and the other two $\mathit{Re}_{D}$ cases. The number of vortices with properly closed streamlines that were identified simultaneously on each snapshot ranges between one and five (figure 15 a). The primary vortex was present in all frames, in agreement with the results reported by Agui & Andreopoulos (Reference Agui and Andreopoulos1992) for a flow with $\mathit{Re}_{D}=10\times 10^{4}$ . Except for the well-defined topologies mentioned previously, however, smaller structures appear intermittently on the flow field and upstream of the HV1 (figure 15 b). Their manifestations in the form of semi-closed streamlines are particularly short-lived. Due to their fleeting nature, it is not possible to draw concrete conclusions about their origin or their final destination. Sometimes, they appear to merge with larger vortices. In other instances, their abrupt disappearance from the imaged flow plane could be attributed to the action of cross-stream pressure gradients. In any case, these transient flow patterns were observed only in the results of the HRe experiment.

Figure 15. (a) Instantaneous flow topology with five vortices and (b) a single well-defined vortex with properly closed streamlines (HV1) and smaller intermittent structures upstream.

Another feature identified exclusively in the cinematography of this experiment refers to the primary vortex. In three flow episodes, an unusually rapid upstream motion of HV1, away from the cylinder, was observed. This feature originated from a mechanism common in every episode: an influx of momentum from the boundary layer feeding HV1 directly (figure 16 a). The incoming fluid patch is trapped within the left and upper periphery of the vortex. In contrast to multiple other similar interactions between boundary layer flow and HV1, the decisive parameter here is the combination of high momentum and low vorticity of the patch. The vortex engulfs this fluid parcel, but fails to impart rotational motion on it. The parcel completes half a revolution within the vortex core until its direction of motion is reversed (figure 16 b). From that point on, the linear momentum of the fluid patch is superimposed on the rotational motion of the vortex. The composite flow structure is advected rapidly to the upstream, riding on top of the near-wall jet of reverse flow (figure 16 c). This migration is eventually impeded either by the emergence of a tertiary vortex at the left toe of HV1, or by the collision with incoming flow (figure 16 d). In both cases, the momentum of fluid masses comprising the vortex is dissipated and flow topology assumes again a more conventional expression. Throughout this episode, the large velocity gradients developing close to the wall change rapidly in space and time. In total, all three episodes characterized by the swift motion of HV1 to the upstream, lasted for 322 ms with a frequency of appearance equal to 0.92 Hz.

Figure 16. Episode of rapid advection of the primary vortex to the upstream. Contour maps are flooded with the non-dimensionalized velocity magnitude, $\Vert U\Vert$ . (a)  $t^{\ast }=8.31$ , (b)  $t^{\ast }=8.42$ , (c)  $t^{\ast }=8.60$ , (d)  $t^{\ast }=8.87$ .

Figure 17. (a) Pulse of high momentum approaching the face of the cylinder and (b) backflow mode. Contour maps are flooded with (a) the non-dimensionalized velocity magnitude, $\Vert U\Vert$ and (b) the streamwise velocity component. (a)  $t^{\ast }=2.14$ , (b)  $t^{\ast }=2.21$ .

Figure 18. Sequence of snapshots illustrating the eruptions of near-wall fluid for the HRe case. Flood contour maps express the magnitude of the out-of-plane vorticity, ${\it\omega}_{z}$ . (a)  $t^{\ast }=5.94$ , (b)  $t^{\ast }=6.03$ , (c)  $t^{\ast }=6.15$ , (d)  $t^{\ast }=6.34$ .

Additional interactions of the horseshoe vortex system with other flow components and the wall also constitute integral parts of the junction flow mechanism for HRe. In general, many of the characteristics that were described in previous cases (LRe, MRe) apply here too and need not be repeated. An important difference, however, is noticed with respect to the elevated levels of momentum that characterize the horizontal boundary layer flow. Large-scale pulses of high velocity penetrate deeper into the flow stagnation region (figure 17 a). They get very close to the face of the cylinder; however, (and unlike what was reported for the MRe case) they cannot establish a downward jet of high momentum. These flow elements energize the general downwash flow, the reverse near-wall jet and occasionally the primary vortex (as described previously). A stronger return flow, which at most times assumes velocities as high as $0.6U_{0}$ , signals the transition to the backflow mode (figure 17 b). This becomes possible only when the incoming near-wall flow becomes weaker due to a reduction of its momentum. It is therefore demonstrated that fluctuations in momentum of the incoming boundary layer indeed influence the switching of the flow between its modes (Devenport & Simpson Reference Devenport and Simpson1990).

In addition to high momentum, the return flow for the HRe case is rich in vorticity. The presence of the layer of positive vorticity (evident in figure 4 c) is continuous. Regardless of the flow mode, this feature is displayed in every snapshot of the flow cinematography. Video animations also reveal that the return flow is fed with positive vorticity emanating from the interaction of the downwash flow with the cylinder. The connection between these two flow components is intermittent and this explains why it is not manifested in the time-averaged results (figure 4 c). Another feature that is brought to prominence through time-resolved analysis is the vortical content of the approaching boundary layer. Multiple patches of vorticity comprise the incoming flow. Negative vorticity dominates the separated flow close to the wall and feeds HV1 and HV2. By contrast, the content of the mid and upper parts of the flow domain is mixed, with negative and positive vorticity clusters alternating in a seemingly random manner. The increased momentum and vorticity levels justify the elevated intensity of eruptions of near-wall fluid (figure 18 a–d) observed exclusively for this HRe experiment. The sequence of snapshots in figure 18(a–d) describes the ejection of near-wall vorticity that occurs simultaneously with the disintegration of the pre-existing HV3. As the bottom of the primary vortex approaches the wall, it displaces fluid masses of lower momentum (figure 18 a). HV3 blocks the horizontal motion of the displaced fluid and directs it upwards at an oblique angle with respect to the wall. This forms a jet characterized by high momentum, which was imparted by the horseshoe vortex and possibly augmented by an instantaneous increment in the velocity of the return flow (figure 18 b). The violent collision of this jet with HV3 causes the disintegration of the latter (figure 18 c). Remnants of the coherent vorticity of the tertiary vortex in the form of small packets are trapped in the rotational motion of the primary vortex. Some of them are diffused within the surrounding flow. The rest complete a full revolution and join again the horizontal layer of positive vorticity that characterizes the return flow (figure 18 d). The whole process repeats at irregular time intervals.

Finally, we refer to the statistics describing the modal dynamics of the junction flow for the HRe case. Again, all three modes mentioned in the analysis of the LRe and MRe cases are present here too. Quantitatively, the HRe case is at zeroflow mode 31.3 % of time. The percentage for intermediate mode is 39.6 % and for the backflow mode is 19.3 %. For the remaining 9.8 % of time, episodes characterized by the rapid motion of the primary vortex to the upstream dominate the flow dynamics. The calculation of the average dimensionless frequencies yielded the following results: 0.69 for zeroflow, 0.77 for intermediate and 0.40 for backflow mode. The frequencies observed at this high level of Reynolds number are lower than those recorded for the other two experiments. Furthermore, and in agreement with the MRe case, the intermediate flow mode is the prevalent one here.

3.3.1. Spatial and temporal dynamics of primary horseshoe vortex

Two-dimensional p.d.f.s of the location of the primary horseshoe vortex complement the understanding of the complex dynamics of junction flows. They provide a concise way to map the topology of this major flow component simultaneously in time and space. The exact position of the vortex core is identified using the so-called $Q$ -criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). The quantity $Q$ is defined as: $Q=({\it\bf\Omega}^{2}-\boldsymbol{S}^{2})/2$ , where ${\it\bf\Omega}$ is the rate of rotation and $\boldsymbol{S}$ is the rate of flow strain. When $Q$ assumes positive values, rotation dominates. The higher the value of $Q$ at specific locations of the flow field, the more confidence there is for the appearance of a coherent vortical structure at these locations. Here, we assume that the position of the primary horseshoe vortex coincides with that of $\mathit{maxQ}$ for every flow snapshot (we excluded from this calculation instants for the LRe case where HV1 was not present).

The outcomes of the $Q$ -criterion analysis are mapped on the two-dimensional flow domains for each $\mathit{Re}_{D}$ case in figure 19(a–c). Regardless of $\mathit{Re}_{D}$ , the orbits of the HV1 core form shapes with roughly elliptical outlines. The horizontal extents of these topologies are $0.15D$ for LRe, $0.17D$ for MRe and $0.14D$ for HRe. The corresponding vertical extents are $0.07D$ , $0.06D$ and $0.06D$ . Therefore, the horizontal amplitude of wandering is higher for the MRe case and the vertical amplitude is higher for the LRe case. Differences were identified with respect to the magnitude of the p.d.f.s for each case. Note that p.d.f.s are indirect indicators of the amount of time that HV1 spent at each location throughout the duration of each experiment. In particular, the most probable locations for the centre of the primary vortex cover a continuous region for the LRe case (figure 19 a). The shape of the region approximates an ellipse with a major axis of $0.09D$ and a minor axis equal to $0.03D$ . For the MRe case the position of HV1 at most times falls within a smaller region that consists of two peaks (prominent locations). Both peaks have a common ordinate of $0.05D$ . The distances between each of them and the leading edge of the cylinder are $0.21D$ and $0.18D$ . On the other hand, figure 19(c), corresponding to the HRe case, shows multiple locations with high values of p.d.f.s regarding the position of HV1.

Figure 19. Maps of p.d.f.s for the location of the core of HV1 based on the $Q$ -criterion for the identification of the vortex core.

Figure 20. Characteristic plots of p.d.f.s for the non-dimensionalized fluctuations of the streamwise (a,b) and wall-normal (c,d) velocity components. The plots refer to the HRe case and their location in the flow domain is described by the coordinates ( $X/D$ , $Y/D$ ): (a) ( $-0.181$ , 0.002); (b) ( $-0.323$ , 0.038); (c) ( $-0.175$ , 0.014); and (d) ( $-0.160$ , 0.104).

A comparison between the results discussed in this section and those reported in past studies provides an additional perspective regarding the dynamics of the HV1. Agui & Andreopoulos (Reference Agui and Andreopoulos1992) used flow visualizations and wall-pressure measurements to argue that the amplitude of wandering of the primary vortex scales with the thickness of the incoming boundary layer. Although not stated explicitly, it is likely that they referred to wandering that occurred exclusively along the horizontal direction. Our data do not support this statement. The amplitudes of horizontal wandering ( $0.15D$ for LRe, $0.17D$ for MRe and $0.14D$ for HRe) differ significantly from the boundary layer thicknesses, which were respectively $0.90D$ , $0.86D$ and $0.52D$ . A somewhat better agreement was found between the amplitude of wandering along the wall-normal direction ( $0.07D$ for LRe, $0.06D$ for MRe and $0.06D$ for HRe) and the displacement thickness of the boundary layer for each case measured $0.07D$ , $0.09D$ and $0.04D$ . Nevertheless, the discrepancies with respect to the MRe and HRe cases still preclude the deduction of a more general conclusion on this subject. In another study, Sabatino & Smith (Reference Sabatino and Smith2009, p. 3) published a map of p.d.f.s for the location of the primary vortex measured for a junction flow of $\mathit{Re}_{D}=1.9\times 10^{4}$ . The comparison with results from our LRe case with $\mathit{Re}_{D}=2.9\times 10^{4}$ reveals strong similarities regarding the spatial extent of the two topologies. Both studies map a nearly identical region within which the core of the primary vortex wanders. Nevertheless, Sabatino & Smith (Reference Sabatino and Smith2009) highlighted the existence of two prominent positions for the location of the vortex (instead of a single continuous region reported here). This discrepancy is likely due to the different numerical levels that were selected to plot the p.d.f. contours.

3.3.2. Bimodality of velocity p.d.f.s

Probability density functions of flow velocities have been widely used as a rigorous means to establish the bimodal dynamics of the horseshoe vortex system (Paik et al. Reference Paik, Escauriaza and Sotiropoulos2007; Kirkil & Constantinescu Reference Kirkil and Constantinescu2009). Here, we follow the methodology introduced by Devenport & Simpson (Reference Devenport and Simpson1990). Namely, we plot p.d.f.s of velocity fluctuations non-dimensionalized by their root-mean-square value. The position of zero instantaneous velocity is marked on the horizontal axis with an asterisk. Without this important detail, it is not possible to establish meaningful links between the p.d.f.s peaks and the modes of interest (zeroflow, backflow and intermediate flow). That is, whether the flow is actually reversed and directed away from the cylinder face depends upon whether the instantaneous velocity (mean plus fluctuation) is negative, zero, or positive. For clarity, in figure 20 an asterisk is used along the respective axes to indicate the value of the dimensionless velocity fluctuation corresponding to zero instantaneous velocity in the indicated coordinate direction. Figure 20(a–d) provides additional insight on this statement. In particular, both figure 20(a,b) display p.d.f.s of streamwise velocity fluctuations with two distinct peaks. Nevertheless, only figure 20(a) manifests the zeroflow and backflow modes that characterize the behaviour of the primary horseshoe vortex. There, the first peak centres approximately on zero instantaneous velocity, as marked on the figure by the asterisk. The second is located at a region where instantaneous velocities are negative, which indicates consistency with backflow mode. In contrast, the two peaks shown on subplot (b) are both in regions of positive instantaneous velocities. Whereas the behaviour of flow at this location is also bimodal, the bimodality is not to be interpreted as a manifestation of the zeroflow–backflow phenomenon. Similar findings are reported from the statistical analysis of the instantaneous wall-normal velocity signals. The typical behaviour of the zeroflow–backflow mode mechanism is evidenced in plots with characteristics similar to figure 20(c). When backflow mode is on, the reverse flow jet is nearly parallel to the bed, so the wall-normal velocities are close to zero (left peak). Once zeroflow mode kicks in, wall-normal velocities assume positive values and result in another peak emerging on the right of the first one. In contrast, figure 20(d) displays a double-peak structure with both peaks located at areas of negative wall-normal velocities. Again, these two peaks are not linked to the bimodal dynamics of the primary vortex, at least not in a way similar to that described by Devenport & Simpson (Reference Devenport and Simpson1990). As far as the intermediate flow mode is concerned, its expression on the p.d.f. plots is not obvious. It could be postulated that it corresponds to those parts of the graphs not belonging to the peaks. Alternatively, the intermediate mode may be prevalent at locations where the p.d.f.s do not exhibit prominent peaks. In any case, subsequent analysis will focus on the dynamics of the two extreme modes (zeroflow, backflow) that have been shown to significantly affect junction-flow-related phenomena (Agui & Andreopoulos Reference Agui and Andreopoulos1992; Ölçmen & Simpson Reference Ölçmen and Simpson1994; Praisner et al. Reference Praisner, Seal, Takmaz and Smith1997; Kirkil & Constantinescu Reference Kirkil and Constantinescu2009; Escauriaza & Sotiropoulos Reference Escauriaza and Sotiropoulos2011a ,Reference Escauriaza and Sotiropoulos b ).

Having revisited the interpretation and the characteristics of double-peaked p.d.f.s within the frame of horseshoe vortex dynamics, we now focus on the effect of $\mathit{Re}_{D}$ . Figure 21(a–c) shows the exact locations in the flow domains where the p.d.f.s exhibit bimodal features related to the zeroflow–backflow phenomenon. Bimodality of streamwise velocity fluctuations is exhibited primarily below HV1. The unsteady regions coincide with the path of the near-wall jet of reversed fluid. Of interest is the orientation of these regions. While for LRe and MRe they are nearly parallel to the bed, for the HRe case the orientation is oblique. This is reminiscent of the characteristic pattern observed in the time-averaged maps of vorticity (figure 4 a–c), TKE (figure 5 a–c) and streamwise turbulence intensity (figure 6 a–c). Note, also, that for the MRe and HRe cases a second bimodal region exists in the area of action of the secondary vortex. This finding underscores that HV2 assumes a more active role within higher $\mathit{Re}_{D}$ flows. The extent of bimodal regions for the wall-normal velocity component is, on the other hand, smaller than that of the streamwise velocity component for the LRe case (figure 21 a). Bimodality of wall-normal velocities occupies larger areas for the MRe and HRe cases (figure 21 a,b). These areas coincide with the core and the outer edges of the time-averaged primary vortex. A notable exception occurs for MRe. For this case, a second bimodal region exists just upstream of HV2. It is possible though, that the existence of this region is also attributed to the behaviour of HV1. As demonstrated both in § 3.2.2 and in figure 21(a–c), the orbiting motion of HV1 envelops a larger area, which stretches beyond $0.3D$ upstream of the origin. When at its remote upstream position, the left edge of the primary vortex overlaps with the bimodal region under consideration.

Figure 21. Bimodal regions of the streamwise (solid grey areas) and wall-normal (areas bounded by black lines) fluctuating velocity components. Time-averaged streamlines are superimposed.

Devenport & Simpson (Reference Devenport and Simpson1990, p. 31) were the only ones that documented in detail bimodal regions for the streamwise and wall-normal velocity components. We attempt here a comparison between their maps and those of the HRe case. Notwithstanding existing differences between the two studies (working fluid, boundary layer thickness, shape of obstacle, measuring technique), the levels of the main parameter of interest, namely the Reynolds number, render this comparison meaningful. According to their results, bimodal p.d.f.s of streamwise velocity characterized a continuous region. Its shape was nearly elliptical and was bounded between $-0.13<X/D<0.37$ in the horizontal direction and between $0.00<Y/D<0.09$ in the vertical direction. Similarly, the area of bimodal p.d.f.s for wall-normal velocities occupied a continuous elliptic-shaped region. Slight differences were reported regarding its extent: the horizontal limits were $-0.12<X/D<0.36$ and the vertical limits were $0.00<Y/D<0.10$ . These numbers indicate a significant overlap between the bimodal regions of each velocity component. Our analysis does not corroborate this feature (figure 21 a–c). Additional discrepancies exist with respect to the overall extent of the bimodal regions. According to the PIV data presented here, the bimodality is exhibited at smaller and non-continuous areas of the flow domain. These areas, however, fall within the limits reported in the aforementioned study. It is therefore concluded that the findings of the two studies are not contradictory. Our results can be regarded as a refinement in the spatial extent of the bimodal regions of junction flows that complements the original findings of Devenport & Simpson (Reference Devenport and Simpson1990).

3.3.3. Flow unsteadiness and velocity p.d.f.s

To further investigate junction flow unsteadiness, we examine the p.d.f.s of velocity fluctuations along vertical transits of the flow field. Five locations are selected. These are representative of the following flow features: (a) incoming boundary layer flow, (b) secondary vortex, (c) saddle region in between HV1 and HV2, (d) primary vortex and (e) downwash/return flow. Figure 22 depicts the exact locations for the HRe case. As underscored previously, the time-averaged flow topologies were not identical for the three $\mathit{Re}_{D}$ experiments. Therefore, to render comparisons between cases meaningful, we ensured that for LRe and MRe, transits b and d crossed the core of HV2 and HV1 respectively. Then, the locations a, c and e were selected to have the same dimensionless distances upstream or downstream of b and d as those shown in figure 22 for the HRe case.

Figure 22. Locations of vertical transits along which instantaneous velocities were extracted and p.d.f.s were calculated.

Figures 23(a–e)–25(a–e) depict the variation of the p.d.f.s of streamwise velocity fluctuations along the aforementioned transits and for each $\mathit{Re}_{D}$ case. Solid black lines correspond to the time-averaged velocities. For the incoming boundary layer flow (subfigures a), the velocity profile overlaps with locations where the p.d.f.s obtain their highest values. The width of the coloured band is approximately 0.5 dimensionless units of $U/U_{0}$ and remains relatively constant along the transit for all three cases. Therefore, the incoming flow is characterized by relatively low velocity fluctuation levels. This pattern, however, is not observed within the system of horseshoe vortices. For example, note that the width of the p.d.f. contour plots increases significantly for transit b of figure 25. The increase takes place within and above the time-averaged footprint of the secondary horseshoe vortex. The same trend characterizes the p.d.f.s of the transits c, which are located in the saddle region in between the two vortices. The phenomenon is particularly pronounced for the LRe case (figure 23 a–e). A wide range of values characterizes the instantaneous velocities at vertical locations from near the bed, up to $0.12D$ . A closer inspection of subfigures (c) also reveals discrepancies between most probable and time-averaged velocities. This feature is most evident in the near-wall regions. There, the no-slip condition results in the retardation of flow with velocities obtaining close-to-zero magnitudes. In this region, time-averaged velocities do not necessarily coincide with the most probable velocity values, because their estimated values become sensitive to the wide range of positive and negative fluctuations of the flow. In this way, it is demonstrated that mean streamwise velocities in a time-averaged sense are not always good estimators of the flow field in the particular regions of the juncture. Furthermore, wide-banded p.d.f.s and strong inflection points in the velocity profiles characterize transits d, which cross the core of the primary vortex. These features are common for every $\mathit{Re}_{D}$ case examined here. Overall, a junction flow consistently becomes more turbulent at the saddle region and at the area of action of the primary vortex. Subfigures (e) verify this conclusion. With the exception of LRe case, the widths of the p.d.f. contours in subfigures (e) are of the same order as those of subfigures (a) and (b). Note also that the p.d.f. contours within the same subplot are skewed negatively, suggesting in this way a departure of the streamwise velocity signal from the normal distribution.

Figure 23. Mean streamwise velocity profiles (black lines) superimposed on p.d.f. contours for the LRe case.

Figure 24. Mean streamwise velocity profiles (black lines) superimposed on p.d.f. contours for the MRe case.

Figure 25. Mean streamwise velocity profiles (black lines) superimposed on p.d.f. contours for the HRe case.