3.1.1. Surface flow visualizations

As stated, initial surface flow visualizations were conducted over the low aspect ratio pin immersed in a laminar cross-flow to identify the separation lines, and thus, the presence of vortical flow features near the surface. Figure 3 reveals the steady-state images associated with each tested pin, where figure 3(a) shows the case where $AR_{p} = 0.25$ and $h/\delta = 0.5$, figure 3(b) represents the case where $AR_{p} = 0.5, h/\delta = 1$ and figure 3(c) corresponds to the case where $AR_{p} = 0.75, h/\delta = 1.5$. Note that these images reveal a top view perspective and the flat plate was placed in the upright position for the purposes of this study. The dark and light areas are associated with regions of high and low shear stress, respectively.

Figure 3.

Top view images of steady-state surface oil flow visualizations for varied geometric conditions, (a) $AR_{p} = 0.25$, $h/\delta = 0.5$, (b) $AR_{p} = 0.5$, $h/\delta = 1$ and (c) $AR_{p} = 0.75$, $h/\delta = 1.5$. The numerical annotations included in each image represent the following: (1) high shear stress region associated with the roll-up of the horseshoe vortex system upstream of the pin, (2) cellular mushroom-type vortex over the pin's free end and (3) accumulated pockets of oil in the wake due to the recirculating arch vortex. In all cases, $Re_{D} = 8.1 \times 10^{3}$ and $Re_{\delta } = 4.05 \times 10^{3}$.

Three primary vortical flow features are discussed herein. The first feature is represented by the separation lines upstream of the pin, known as the horseshoe vortex (HV). These result from the spanwise vorticity, $\Omega _z$, associated with the approaching laminar boundary layer, which is caused to separate under the local adverse pressure gradient. Subsequently, the boundary layer rolls up and around the base of the pin to form a system of counter-rotating, streamwise-oriented vortices. The second vortical flow feature is a resultant of the characteristic three-dimensional separation over the pin's free end, denoted by the outlined circumferential edge (in white). The three-dimensional separation bubble imposes a cellular ‘mushroom-type’ or ‘tornado-like’ vortex (Roh & Park Reference Roh and Park2003), which will be discussed in further detail in § 3.1.2. The variation in separation behaviours as a function of $AR_{p}$ and $h/\delta$ is predominantly dictated by the variation in the shear strength of the incoming flow over the pin. As the pin becomes less submerged within the boundary layer, (i.e. $h/\delta$ and $AR_{p}$ both increase), the free end is exposed to higher momentum flow, producing a larger magnitude of shear over the top, with a more defined mean reattachment line, as observed in figure 3(c). The third vortical feature of interest is observed in the pin's near wake, where the pattern of accumulated oil is suggestive of the presence of the base ‘legs’ of the arch vortex.

Based on the observations from figure 3, it is understood that as the height of the pin is increased (as is $AR_{p}$), the location of boundary layer separation moves upstream due to the increased local adverse pressure gradient (i.e. increased blockage). Moreover, in accordance with previous work by Sumner (Reference Sumner2013), the size of the separation bubble and location of reattachment are pushed towards the circumferential trailing edge as the pin's height aspect ratio increases. Under the present test conditions, the effects of aspect ratio and relative submergence are varied concurrently, both contributing to the nature of the three-dimensional separation over the pin's free end.

For figure 3(a), the pin is deeply submerged within the laminar boundary layer; thus, the separation bubble is small and the reattachment point occurs very close to the circumferential leading edge. This can be inferred from the lack of oil present over the majority of the free end surface, representative of the high shear associated with attached flow.

Although similar behaviour is observed in figure 3(b), it is believed that the oil does not track exactly with the flow, and therefore does not accurately depict the mean flow details for the case of $h/\delta =1$ ($AR_{p}=0.25$) due to relatively weak shear. This claim is offered based on the SPIV results, discussed in § 3.1.2, which demonstrate a three-dimensional vortex over the pin's free end. However, this vortical flow feature is likely too weak to be revealed by the oil mixture used in these experiments.

In contrast, figure 3(c) reveals a distinctive cellular vortex pattern over the pin's free end as the height is increased to $h/\delta = 1.5$. This is a result of increased shear and separation angle as the outer flow encounters the tip of the pin, demonstrating more distinctive separation lines over the free end. The cellular separation lines delineate the mushroom-type vortex, which has been observed in previous studies of finite-span cylinders (Roh & Park Reference Roh and Park2003). Note that in all cases shown here, there is a thin layer of accumulated oil along the circumferential trailing edge of the pin as a result of the surface tension between the oil and the pin's top surface. In the near wake, there are two pockets of accumulated oil near the pin's base, accompanied by a separation line along the centre, revealing a symmetric wake structure. As will be discussed in § 3.1.2, the regions of accumulated oil are associated with the ‘legs’ of the mean arch vortex, which pulls fluid from the outer flow in towards the centre wake. The relative size of the arch vortex scales with $AR_{p}$ (and $h/\delta$), as shown by the increased accumulation of oil and spanwise spread. The arch vortex induces a large spanwise flow and downwash velocity in the near wake. Accordingly, the higher momentum flow in the centre induces the formation of two counter-rotating trailing vortices, resulting in a separation line along the symmetry plane.

An inextricable link exists between the flow behaviour over the pin's free end and in the near wake. The relationship between these vortical flow features is defined in terms of the nature of the 3-D separation bubble, the width of the wake and the structure of the recirculation region (i.e. symmetric arch vortex). Although this observation is initially qualitative, these surface flow patterns allowed for an initial assessment of the sensitivity of the vortex dynamics to the geometric parameters under investigation. The assertion regarding the interplay of vortex dynamics between the pin's free end flow and the near wake is quantitatively supported from SPIV measurements discussed herein.

3.1.2. Three-dimensional flow field measurements using SPIV

In order to properly resolve the size and scale of the constituent vortical flow features, the region of interest was sub-divided into three planes: (i) upstream of the pin, to capture the approaching flow, (ii) over the pin's free end and (iii) the near wake region. Initially, the vortex dynamics is discussed for laminar boundary layer conditions (i.e. $Re_{x} = 6 \times 10^{5}, \delta = 6$ mm). The upstream flow is quantified along the symmetry plane and shown in figure 4 for each case as a function of the $AR_{p}$ and $h/\delta$. Here, streamlines of the mean flow are presented for each case in figures 4(a)–4(c), respectively. Corresponding time-averaged streamwise (figures 4d–4f) and wall-normal (figures 4g–4i) velocity profiles are provided at several streamwise locations to further characterize the formation, size, and locations of influence of the HV system. It is noted that, for all cases, the pin's aspect ratio is considered very small, i.e. $AR_{p} < 1$, with a constant diameter-based Reynolds number. There are several important dimensionless parameters that characterize the class of the HV system, including the streamwise distance from the centre of the pin to the primary vortex core, $x_{v}/D$, the distance to the point of upstream boundary layer separation, $x_{s}/D$, and in some cases the regular oscillatory behaviour, $St$. Some of these parameters are referred to herein as a means to evaluate the physical mechanisms associated with the characteristic changes to the HV system.

Figure 4.

Time-averaged streamlines along the centreline plane upstream of the pin for: (a) $AR_{p} = 0.25$, $h/\delta = 0.5$, (b) $AR_{p} = 0.5$, $h/\delta = 1$ and (c) $AR_{p} = 0.75$, $h/\delta = 1.5$. Note that a small portion of the windward side of the pin is included in each plot. The corresponding time-averaged streamwise and wall-normal velocity profiles for several streamwise positions are provided in (d–f) and (g–i), respectively. $Re_{D} = 8.1 \times 10^{3}$.

As can be observed from the time-averaged streamlines (figures 4a–4c), the area of influence of the formation of the system of vortices extends farther upstream and away from the wall as the pin's relative height is increased. This is a direct consequence of the increased local adverse pressure gradient (i.e. increased blockage), which coincides with the point of boundary layer separation being pushed upstream. This earlier onset of boundary layer separation induces spatial growth of the HV system by diffusion. Although the inception and spatial extent of the HV system vary from case to case, the classification of the vortical flow regime is dependent upon $Re_{D}$ and $D/\delta$, which both remain fixed here. As was proposed by Baker (Reference Baker1979), the need for the velocity gradients to be continuous implies the presence of mirrored vortices, which are counter-rotational to the three dominant clockwise rotating vortices observed in the streamline plots. Under this assumption, the HV systems presently studied are categorized as a ‘regime 5’ vortex flow with a total of six vortices. Note that due to limited spatial resolution of the SPIV measurements near the wall, the mirrored, counter-rotating vortices, which are much smaller (i.e. ${<}1$ mm) are not captured. It is understood that for finite-span cylinders, the number of vortices comprised in the laminar HV system will increase as a function of $Re_{D}$ and $D/\delta$, and experiences unsteady behaviours beyond a critical threshold. As mentioned, both parameters are fixed in these experiments and thus, the HV system consists of the same number of vortices.

The corresponding streamwise velocity profiles are provided in figures 4(d)–4(f) at several streamwise locations to distinguish the origins and development of the vortical system in the range $-2 \leq x/D \leq -0.75$. Note that the front face of the pin is located at $x/D = -0.5$. For the shortest pin, $h/\delta = 0.5$ ($AR_{p} = 0.25$, figure 4d), the velocity profiles farthest upstream, $x/D = -2$ (diamonds), reveal a small influence of the presence of the pin. As distance from the pin is reduced, the adverse pressure gradient overcomes the inertial forces, inducing boundary layer separation, defined at $x_{s}/D \approx -1.5$. At locations closer to the pin, i.e. $x/D \geq -1$, the profiles reveal reverse flow near the wall due to the presence of the HV (see figure 4a). The mean position of the primary HV (noted as I) is located at $x_{v}/D \approx -0.6$, $y/D \approx 0.1$. The mean secondary and tertiary vortices (i.e. II and III, respectively) are fed from the vorticity of the primary vortex and remain closer to the wall. This is due to the induced velocity, which entrains the fluid downward. The time-averaged positions of the secondary and tertiary vortices are located at $x_{v}/D \approx -0.8$, $y/D \approx 0.08$ and $x_{v}/D \approx -0.96$, $y/D \approx 0.07$, respectively.

As the relative height of the pin is increased, $h/\delta = 1$ ($AR_{p} = 0.5$, figure 4e), there is an upstream shift in the location of boundary layer separation and the position of the mean primary HV core, as anticipated. Boundary layer separation occurs approximately 25 %–30 % further upstream, i.e. $x_{s}/D \approx -1.95$, and the position of the primary HV core is shifted to $x_{v}/D \approx -0.75$, $y/D \approx 0.13$, respectively. Accordingly, the positions of the secondary and tertiary vortices are shifted to $x_{v}/D \approx -1.1$, $y/D \approx 0.1$ and $x_{v}/D \approx -1.34$, $y/D \approx 0.088$, respectively. These positions are lifted from the surface by approximately 25 %–30 %, relative to the case for $h/\delta = 0.5$. This is clearly observed by the increased momentum deficit, which is present up to a wall-normal distance of $y/\delta \approx 1.75$.

As expected, when the pin's height is further increased to $h/\delta = 1.5$ ($AR_{p} = 0.75$, figure 4f), the momentum deficit is increased, influencing the flow farther upstream. At the most upstream position shown, the boundary layer has already separated, and was determined to be $x_{s}/D \approx -2.13$ (i.e. 9 % farther upstream than the previous case). Consequently, the average positions of each of the vortex cores are located farther upstream, where the primary vortex core is located at $x_{v}/D \approx -0.81$, $y/D \approx 0.14$. The secondary and tertiary vortices are located at $x_{v}/D \approx -1.2$, $y/D \approx 0.1$ and $x_{v}/D \approx -1.5$, $y/D \approx 0.094$, respectively. As stated, the system of vortices is uniformly shifted farther upstream from the pin as a function of the pin's increasing height (i.e. both $AR_{p}$ and $h/\delta$ increase), synonymous with an increased local adverse pressure gradient, which promotes earlier boundary layer separation and onset of the HV system.

The corresponding time-averaged, wall-normal velocity profiles are shown in figures 4(g)–4(i). In a similar fashion, as the distance from the pin is reduced, the influence from the vortical structures becomes more prominent. Likewise, this trend is also observed as a function of increased $h/\delta$ (and $AR_{p}$). For the smallest pin (figure 4g), the location measured nearest to the body is in between the primary and secondary vortices, thus revealing downwash associated with a portion of the secondary vortex as it encounters the wall. Farther from the pin, the mean profiles reveal reduced impact from the HV system. As the pin's height is increased, the influence of the HV is felt farther upstream, demonstrated by the increased magnitude of the undulations in the profiles, specifically for $x/D \geq -1$ (figure 4i). Moreover, at $x/D = -0.75$ (inverted triangles), the locations of inflection in $v/U_{\infty }$ demonstrate the spatial growth of the vortices as that point moves farther from the wall with increased pin height (i.e. increased $h/\delta$ and $AR_{p}$).

The three-dimensional flow field characteristics inherent to this configuration are explored in further details at the SPIV window 2 location (see figure 1). Here, the field of view captures the mean flow features over the pin's free end and the near wake. Within this region, a pair of counter-rotating streamwise vortices develop near the free end as a result of the interaction between upward-directed separated flow around the sides of the cylinder and the downward-directed flow over the top. This behaviour induces counter-rotating vortices, which roll over the top edges, forming the tip vortices (e.g. Kawamura et al. Reference Kawamura, Hiwada, Hibino, Mabuchi and Kumada1984). As the tip vortices extend in the streamwise direction, they induce a downwash along the wake centreline and evolve into the trailing vortex pair. These secondary interactions are thought to be the mechanism by which the trailing vortex pair is formed and evolve into a quasi-steady, counter-rotating pair near the wall along the outside of the pin's wake. The tip vortices are thought to be relatively short-lived structures with an extension length near $x/D \leq 2$ (e.g. Zhu et al. Reference Zhu, Wang, Wang and Wang2017), which extends to the end of the measurement domain in the present work. Although there is some discrepancy on the exact definition of the tip and trailing vortical flow features, the authors believe these distinctive features to be mutually inclusive for wall-mounted short aspect ratio cylinders. Each of these features and the mechanisms of their formation and evolution are discussed herein.

The mean flow fields of streamwise velocity and corresponding spanwise vorticity, $\Omega _{z}D/U_{\infty }$, are presented along the pin's symmetry plane in figure 5 to emphasize the characteristics of the separation bubble, wake recirculation and separating shear layer. As observed in figures 5(a) and 5(d), the effect from the presence of the pin on the outer free-stream flow is very small due to its full submergence within the boundary layer. Accordingly, a very small separation bubble is formed over the free end. Due to the nature of the interaction between the incoming cross-flow with the pin's sharp leading edge, the flow must experience a small separation bubble near the leading edge; however, it is not captured with the current measurement resolution. However, the small separation may be inferred from the contours of spanwise vorticity. Figure 5(d) reveals the largest magnitude of negative spanwise vorticity near the leading edge, suggesting high shear and the presence of a small separation bubble, which reattaches near $x_{r} = 0.16D$. Note that $x_{r}$ originates at the pin's circumferential leading edge and refers to the reattachment point over the top, as defined by the location at which the streamwise velocity changes sign. The concentration of spanwise vorticity near the top surface of the pin further suggests that the flow remains attached for the majority of the surface. As such, the flow experiences a very shallow angle of separation from the pin's trailing edge, and reattaches in the wake near $x/D = 1.34$.

Figure 5.

(a–c) Colour contours of time-averaged streamwise velocity, $u/U_{\infty }$, along the centreline plane, superimposed with in-plane streamlines, and (d–f) corresponding contours of spanwise vorticity, $\Omega _{z}D/U_{\infty }$. Each row represents a given geometric test case, such that (a,d) $AR_{p} = 0.25$, $h/\delta = 0.5$; (b,e) $AR_{p} = 0.5$, $h/\delta = 1$; (c,f) $AR_{p} = 0.75$, $h/\delta = 1.5$. $Re_{D} = 8.1 \times 10^{3}$.

As the pin's relative height is increased, the separation bubble grows significantly. In the case for $h/\delta =1$ (figures 5b and 5e), a recirculation zone extends from the leading edge to $x_{r} = 0.81D$. Whereas, for $h/\delta = 1.5$ (figures 5c and 5f), the mean separation bubble reattaches near $x_{r} = 0.87D$. The increased size of the separation bubble and steeper angle of flow separation from the trailing edge of the pin are a result of increased exposure of the pin to higher momentum flow. As the height of the pin extends beyond the local boundary layer thickness, the magnitude of upward-directed flow, tangential to the pin's front face and around the sides, increases. These behaviours will be further corroborated in figure 7.

The three-dimensional flow over the free end surface has been shown to be an integral element in the determination of near wake behaviour for a finite-span pin. A general overview of the local flow characteristics over the free end (using quantitative measurements from the present study) is presented in figure 6(a) and compared with qualitative work by Roh & Park (Reference Roh and Park2001, Reference Roh and Park2003) in figure 6(b). The referenced work conducted flow visualizations over a finite-span cylinder with an aspect ratio of 1.25 for a similar Reynolds number. The top image in figure 6(a) represents an orthogonal view of the streamlines measured just above the free surface, at ${\rm \Delta} y = 0.01D$, for the case of $AR_{p} = 0.5$ (and $h/\delta = 1$). It is important to note that the influence of the local boundary layer was considered negligible in the referenced work (figure 6b), i.e. $h/\delta \approx 20$, and thus, some differences may be attributed to the coupled nature of $AR_{p}$ and $h/\delta$, which are both $O(1)$ in the present study.

Figure 6.

Local flow field surface topology shown for (a) measured time-averaged streamlines above the free end surface for the present case of $AR_{p} = 0.5$, $(h/\delta = 1)$ and $Re_{D} = 8.1 \times 10^{3}$, compared with (b) qualitative experimental results from Roh & Park (Reference Roh and Park2001, Reference Roh and Park2003) where $AR_{p} = 1.25, (h/\delta \approx 20)$ and $Re_{D} = 5.92 \times 10^{3}$ and $1.48 \times 10^{5}$. The label (A) indicates the location of the nodes. F: focus point; S: saddle point; N: node; CW: clockwise rotation; CCW: counterclockwise rotation.

The free end flow is topologically characterized by the presence of two foci, F, in the fore half-portion associated with the cellular mushroom-type vortex (also referred to as ‘eye-like spots’ in Roh & Park (Reference Roh and Park2001, Reference Roh and Park2003)) and two nodes, N, in the rear half from which the two tip vortices originate and subsequently enter the wake. The dashed line along the inner circumference is representative of the separation line, from which two saddle points exist (S) along the symmetry plane, one at the point of separation and the other at reattachment, respectively. A corresponding isometric view of the data collected in the present study is also provided in the bottom of figure 6(a), which includes the vertical symmetry plane above the pin to compare the experimental results to the referenced articles (figure 6b).

The discussion of the three-dimensional flow physics associated with the free surface is further elaborated with a comparison between each tested pin in figure 7. The flow fields are presented at a horizontal (x–z) plane, offset from the surface by ${\rm \Delta} y = 0.01D$ (or $0.02\delta$) for each case. Here, the mean streamwise and wall-normal velocity fields are compared. As stated previously, for $h/\delta =0.5$ ($AR_{p} = 0.25$, figures 7a and 7d), the resolution in the measurements is insufficient to capture a distinctive separation near the free surface. Since the pin is fully submerged in the boundary layer, it experiences more downward directed flow due to the associated spanwise vorticity, such that any upwash flow deflected over the top of the pin is of relatively low magnitude, as shown just upstream of the pin's leading edge (figure 7b). Although not observed here, this results in a very small separation bubble. The location of the separation bubble may be speculated from the slight acceleration (seen from the colour contours) just downstream of the leading edge, followed by a decelerated streamwise velocity over the majority of the surface. Consequently, the first 10 % of the pin's span, $-0.5 \leq x/D \leq -0.4$, corresponds to a very small magnitude of upwash. This region of slight upwash is indicative of the upward deflected flow over the small-scale laminar separation bubble. Furthermore, the presence of the tip vortices may be inferred from the behaviour along the outside edges of the pin ($|z/D| = 0.5$) near the centre span. Here, there is streamline convergence, suggesting an acceleration of the flow as it rolls over the top edges of the pin. Moreover, there are streak patterns of acceleration, coupled with a small region of decelerated flow, which extend longitudinally along the outer edges of the wake. This behaviour is suggestive of the presence of streamwise tip vortices. Consequently, two concentrations of downwash are observed in the wake, figure 7(d), providing further evidence of the induced downwash resulting from the tip vortices. The combined downwash in the mean flow, and mutual inductance of downwash from the tip vortices cause the tip vortices to impinge onto the wall in the near wake. The secondary vorticity generated near the wall ultimately leads to the formation of the trailing vortex pair farther downstream.

Figure 7.

Top view of time-averaged (a–c) streamwise, $u/U_{\infty }$, and (d–f) wall-normal velocity, $v/U_{\infty }$, at a vertical position of ${\rm \Delta} y/D = 0.01$ above the free end surface. Each row represents a given geometric case, such that (a,d) $AR_{p} = 0.25$, $h/\delta = 0.5$; (b,e) $AR_{p} = 0.5$, $h/\delta = 1$; (c,f) $AR_{p} = 0.75$, $h/\delta = 1.5$. For all cases, $Re_{x} = 6 \times 10^{5}$ and $Re_{D} = 8.1 \times 10^{3}$. The numerical annotations represent several flow features: (1) concentration of upwash flow as incoming flow interacts with the front face of the pin and is directed upward and over the top, (2) separation line associated with the free end separation bubble, (3) cellular mushroom-type vortex foci, (4) reattachment point over the pin's free end and (5) region of opposing signs of wall-normal velocity associated with the roll-up of the tip vortices which extend into the wake.

As the pin's height is increased, the separated flow over the top of the pin becomes more prominent, as outlined by the white separation line in figures 7(b) and 7(c). For this coupled change in $h/\delta$ and $AR_{p}$, the free end is characterized by a cellular mushroom-type vortex having symmetric counter-rotating foci with rotational sense in towards the centre of the separation bubble. The foci are located just upstream of the pin's centreline at $x/D \approx -0.08$ and $|z/D| \approx 0.3$ and $0.34$ for $h/\delta =1$ and $1.5$, respectively. The locations of the foci are very similar, but the size and strength of the mushroom vortex scales relative to the increased pin height. Thus, as the pin height increases to $h/\delta =1.5$ ($AR_{p}=0.75)$, the magnitude and extent of reverse flow is increased, accompanied by stronger shear in the separating vortex sheet over the bubble.

As mentioned previously, the presence of two additional vortical flow features may be inferred from these measurements: the streamwise tip and trailing vortex pairs. This is most clearly observed for the cases where $h/\delta =1$ and 1.5, (figures 7b and 7c, respectively), by the regions of streamwise velocity deficit in the wake, which emanate from the sides near $|z/D| \approx 0.5$. In accordance with this, the wall-normal velocity fields reveal an upwash along the outside edges of the rear portion, accompanied by strong downwash in the central wake. Note that the downwash shown over the top of the pin in the aft portion represents the reattachment and subsequent generation of the tip vortices at $x/D \approx 0.14$ and $0.18$ (figures 7e and 7f, respectively). As the tip vortices extend into the wake and merge with the recirculation region (i.e. arch vortex), the combined actions of these vortical structures induce a stronger downwash flow in the centre wake, entraining higher momentum outer flow towards the wall. The competing spanwise and axial vorticity components associated with the arch and tip vortices initiate the secondary interactions, which lead to the development of the trailing vortex pair. Further description of the formation of these vortical features is discussed in figure 9.

Based on the inter-dependent nature of the vortical features over the pin's free end and those in the wake, the characteristics of the separation bubble are explored as a function of the concurrent changes in aspect ratio and relative submergence within the boundary layer. Figures 8(a) and 8(b) present the locations of reattachment for each case, plotted against a broad spectrum of related work as a function of $AR_{p}$ and $h/\delta$, respectively. The comparison provided in figure 8(a) reveals variations in reattachment as a function of aspect ratio from the review by Sumner (Reference Sumner2013), where the data from the present study have been superimposed and are shown by the circled symbols. Likewise, figure 8(b) demonstrates the variation in reattachment location as a function of $h/\delta$ from the work of Tsutsui & Kawahara (Reference Tsutsui and Kawahara2006). This parametric study was conducted in laminar conditions for aspect ratios of $AR_{p} = 0.25$, 0.5, 0.75 and 1, which corresponded to $h/\delta = 0.5$, 1, 1.5 and 2, respectively.

Figure 8.

(a) Locations of the reattachment saddle point on the centreline of the free end surface for finite-span cylinders as reviewed by Sumner (Reference Sumner2013), compared with data from the present work (circled symbols). All data presented here were collected within a range of $10^{3} \leq Re_{D} \leq 10^{5}$. (b) Variation in the reattachment location as reviewed by Tsutsui & Kawahara (Reference Tsutsui and Kawahara2006) for turbulent boundary layer conditions, i.e. $1.1 \times 10^{4} \leq Re_{D} \leq 1.1 \times 10^{5}$, compared with data from the present work in laminar boundary layer conditions, i.e. $Re_{D} = 8.1 \times 10^{3}$.

From figure 8(a), a discerning relationship between the aspect ratio and location of reattachment can be identified. As the aspect ratio is increased, the reattachment point moves downstream. The data from the present study follow a similar trend to previous researchers, although it is important to note that the parameters of $Re_{D}$ and $h/\delta$ are not matched. In particular, the reattachment point for the $AR_{p} = 0.25$ ($h/\delta =0.5$) is presumably much closer to the circumferential leading edge than has been proposed in previous work. This is likely because in the present case the pin is fully submerged within the boundary layer, resulting in a much smaller region of separated flow and subsequently, an earlier reattachment. The reattachment point identified from the present study occurs slightly farther downstream than those found in previous work. It is important to note that most of the data provided by previous research were collected for low aspect ratio cylinders as they interacted with a tripped or fully turbulent boundary layer flow, and thus, the reattachment point is expected to occur earlier, relative to laminar flow conditions.

To examine the effect of the pin's relative submergence on the reattachment behaviour, the present data were compared with the qualitative oil flow visualization work from Tsutsui & Kawahara (Reference Tsutsui and Kawahara2006) in figure 8(b). They explored the variation in flow reattachment for a similar range of parameters, i.e. $AR_{p} \leq 1$ and $h/\delta \approx O(1)$, although the study was conducted with a turbulent boundary layer. The quantitative data from the present work are designated by circled symbols (figure 8b) and superimposed onto the qualitative results from Tsutsui & Kawahara (Reference Tsutsui and Kawahara2006). Note that the variation in the reattachment locations is a function of the Reynolds number, as that is the only variable between the compared data sets. Similar to the trend shown in figure 8(a), the locations of flow reattachment are in good agreement with that shown from the literature, although they are uniformly shifted slightly farther downstream as a result of the difference in boundary layer regime. Based on the similarities between the data sets, it may be speculated that the weighted contribution from the pin's aspect ratio may be greater than the effect of Reynolds number, and effectively, the parameter of $h/\delta$. This claim will be examined in § 3.2 to explore the effect of $Re_{x}$ and $h/\delta$ for a fixed $AR_{p}$.

The detailed free end flow physics prompts an examination into the wake flow features, thereby shifting focus to SPIV window location 3. As mentioned, the flow features over the top of the pin have direct implications on the characteristic behaviour and vortical interactions associated with the wake. A prominent mushroom-type vortex, or recirculation bubble, was observed over the top of the pin, specifically for the cases of $h/\delta = 1$ and 1.5 (figures 7b and 7c, respectively). As stated, the two foci of the mushroom vortex were demonstrated with their respective sense of surface-normal rotation. Although the flow field here is described in a time-averaged sense, the dynamics between the mushroom-vortex feature and the wake has been documented such that the wall-normal vortices lift off from the free surface through mutual induction and influence of the outer shear layer. As the mushroom vortex lifts away from the surface, the influences of the mean flow and strong spanwise vorticity component cause the axis of the vortex foci to bend into the wake. As a result, a prominent cross-stream vortex is established in the mean recirculation region, herein defined as the arch vortex. Consistent with literature, the symmetric shedding mode (or arch-type shedding) is expected for this range of sub-critical aspect ratio, i.e. $AR_{p} \leq 1$ (Pattenden et al. Reference Pattenden, Turnock and Zhang2005).

The relative size and strength of the vortical features in the wake are described in figure 9 via iso-surfaces of the $Q$-criterion ($Q = 0.25$, 0.5 and 1.5, where $Q$ refers to value(s) calculated using the $Q$-criterion vortex identification method, which identifies regions of the flow where the vorticity magnitude is greater than the magnitude of rate-of-strain), which are coloured by the dimensionless turbulent kinetic energy, ${\rm TKE}/U_{\infty }^{2}$. This representation of the flow helps to convey the unsteadiness associated with these flow features. Characteristic to this type of flow is the M-shaped arch vortex, also reported in Zhu et al. (Reference Zhu, Wang, Wang and Wang2017). For the shortest pin (figure 9a), the fluctuations about the mean are very small, due to the pin's full submergence within the laminar boundary layer and thus, generates a very small recirculation region. As the pin's height is increased, the unsteadiness in the wake grows. The intensity of the fluctuations is largest at the downstream edge of the arch-vortex structure, suggestive of the location at which the arch vortex sheds away from the pin periodically. The $Q$-criterion also reveals the quasi-steady horseshoe vortices along the base of each pin. Additionally, the trailing vortex pair is observed in the central wake region, generated by the combination of induced downwash and secondary vorticity associated with the tip and arch vortices. The largest magnitude of turbulent fluctuations is located within the vortex formation region in the proximity of the oscillating reattachment point in the wake. Correspondingly, the relative magnitude and distribution of the kinetic energy associated with the turbulent fluctuations also increase as the pin's wake becomes more extensive.

Figure 9.

Time-averaged iso-surfaces of $Q$-criterion ($Q = 0.25$, 0.5, and 1.5) coloured by $TKE/U_{\infty }^{2}$ for: (a) $AR_{p} = 0.25$, $h/\delta = 0.5$; (b) $AR_{p} = 0.5$, $h/\delta = 1$; and (c) $AR_{p} = 0.75$, $h/\delta = 1.5$. Here, $Re_{x} = 6 \times 10^{5}$ ($Re_{D} = 8.1 \times 10^{3}$).

Some of the physical mechanisms responsible for the generation and evolution of these vortical flows are described more quantitatively in figure 10, which presents the downstream evolution of the dimensionless circulation per unit area, $\Gamma ^*_A$, and the dimensionless integrated TKE, $TKE^*_{int}$, computed upstream and in the wake for each case. Note that each quantity has been non-dimensionalized by the pin's diameter, D, and the free-stream velocity, $U_{\infty }$. A sample flow field along the centreline for the case of $h/\delta =1$ ($AR_{p} = 0.5$) is included inset for each plot.

Figure 10.

(a and b) Dimensionless circulation per unit area, $\Gamma ^*_A$, compared with (c and d) integrated dimensionless fluctuating energy content, $TKE^*_{int}$, upstream of the pin, (${x/D} \leq -0.5$), and in the wake, ($x/D \geq 0.5$), respectively. Here, the incoming boundary layer is laminar ($Re_{x} = 6 \times 10^{5}$ and $Re_{D} = 8.1 \times 10^{3}$). Representative mean flow fields along the centreline are shown insets for the case of $h/\delta = 1, (AR_{p} = 0.5)$.

The circulation generated (per unit area) associated with each of these cases is demonstrated in figures 10(a) and 10(b). As expected, the magnitude of circulation in the approaching flow grows as a function of increased blockage by the pin (i.e. increased $h/\delta$ and $AR_{p}$). Accordingly, the undulations in the circulation profiles directly correspond to the locations of the horseshoe vortices, which are pushed further upstream as a function of the pin's increasing height. However, the peak in circulation exists nearest the pin's trailing edge, near $x/D \approx 0.5$ (figure 10b), reaching a value greater than the maximum value upstream of the pin. This is evidence of the additional accumulated circulation that has been generated around the pin sides and over the free end. Subsequently, there is a steep drop in circulation, which levels to a fairly constant value in the region of the arch vortex (i.e. $x/D \leq 1$). Moreover, beyond $x/D = 2$, the circulation reaches an asymptotic value due to flow reattachment.

The evolution of $TKE^*_{int}$ is plotted in figures 10(c) and 10(d) with sample contours of $TKE/U_{\infty }^{2}$ included in the insets to distinguish regions of greatest unsteadiness. Upstream of the pin, the profiles reveal maximum unsteadiness in the region between the primary and secondary HV, for cases of $h/\delta = 1$ and 1.5, respectively. This behaviour is due to the momentum transfer, or ‘feeding’, of vorticity and fluctuations between the energy in the larger scales. In contrast, the total fluctuating energy present in the surrounding flow for the pin where $h/\delta = 0.5$ reveals a nearly constant value throughout the measurement domain. This indicates that the pin's deep submergence within the incoming boundary layer offers a minimal impact on the mean flow, acting more closely to that of a roughness element. Whereas, the unsteadiness associated with the HV and recirculation region grows significantly as the pin's height approaches the local boundary layer thickness. The turbulent fluctuations extend much farther downstream for the cases of $h/\delta = 1$ and 1.5, demonstrated by the difference in the magnitude at the final measurement location of $x/D = 3$.