3.1. Time-averaged wake

For all $H/D$ considered, the mean wake flow is characterized by a 3D recirculation, shown in figure 4, which extends to $x/D \approx 4$. Similarly to experimental observations for the square cylinder for $\delta /D \geq 1$ (Wang et al. Reference Wang, Zhou, Chan and Lam2006; Hosseini et al. Reference Hosseini, Bourgeois and Martinuzzi2013), a downwash region forms downstream of the free end and a region of upwash behind the obstacle–ground junction. The counter-rotating free end and junction recirculating regions are centred about foci and bounded by the positive bifurcation lines. These lines extend to a saddle point. The vertical location $y/H$ of the saddle point is only weakly affected by $H/D$, i.e. the saddle location lies between $0.45H$ and $0.5H$ for the cases shown in figure 4. Moreover, the general topology of the wake is similar for all aspect ratios. The width of the wake ($z$ direction, not shown for brevity) remains unchanged at the height of the foci. The positions of the foci relative to the free end and junction in the plane $z=0$ remain similar for all aspect ratios. For $H/D = 3$, a slight reduction in the streamwise extent of the recirculation region from $x/D = 4.1$ to $x/D = 3.9$ can be observed. This is consistent with the observations of Wang & Zhou (Reference Wang and Zhou2009) ($\delta /D = 1.35$, $Re = 9300$) for the square cylinder and Rostamy et al. (Reference Rostamy, Sumner, Bergstrom and Bugg2012) ($\delta /D = 1.6$, $Re = 42\,000$) for the circular cylinder, both studies noting a streamwise reduction in the extent of the recirculation for $H/D = 2$.

Figure 4.

Time-averaged sectional streamlines at $z/D = 0$ for $H/D = 3$, 4, 7: (a) $H/D = 3$, $\delta /D = 0.98$, (b) $H/D = 4$, $\delta /D = 0.98$ and (c) $H/D = 7$, $\delta /D = 0.98$.

To identify vortex cores in the three-dimensional three-component (3D-3C) velocity field and educe time-averaged and instantaneous vortex structures, the Q-criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988) is applied. Isosurfaces of $Q = 0$ are displayed in figure 5 for $H/D = 3$, 4, 7. The figure shows an arch-type structure containing the recirculating region, similar to that shown by Zhu et al. (Reference Zhu, Wang, Wang and Wang2017) for the $H/D = 2$ cantilevered cylinder. Four distinct branches of Q-isosurfaces extend downstream from the arch structure. These streamwise branches are denoted as poles in this study and correspond to time-averaged tip (poles 1, 2) and base (poles 3, 4) vortices. A similar quadrupole topology was observed for $H/D = 5, 6$, and has been reported for finite square cylinders (Hosseini et al. Reference Hosseini, Bourgeois and Martinuzzi2013) for $\delta /D \geq 1$.

Figure 5.

Time-averaged isosurfaces of $Q/(U_{\infty }/D)^{2} = 0$ in grey isosurface: (a) $H/D = 3$, (b) $H/D = 4$ and (c) $H/D = 7$. The arch vortex is shown in red isosurfaces of $Q/(U_{\infty }/D)^{2} = 0.0015$. Isosurfaces of $|\omega _x|= \max (|\omega _x|/2)$ are overlaid in blue.

To investigate the 3D spatial distribution of the time-averaged vorticity field, isosurfaces of $\overline{\omega _x}$, $\overline{\omega _y}$ and $\overline{\omega _z}$ are shown in figure 6 for $H/D = 4, 7$ and $\delta /D = 0.98$. Similar vorticity distributions are observed for all aspect ratios investigated. As expected, figures 6(a) and 6(d) show concentrated regions of high-magnitude $\overline{\omega _y}$ corresponding to the separated shear layers. Figures 6(c) and 6(f) show concentrated regions with large values of $\overline{\omega _z}$ at the cylinder tip and base regions. The negatively signed region of $\overline{\omega _z}$ near the cylinder tip is associated with separation over the free end and is consistent with the location of the bifurcation line along the separated shear layer. The positively signed region of $\overline{\omega _z}$, which forms downstream of the cylinder junction (figure 6c,f), coincides with the region of upwash and cannot be explained simply by the local separation process along the cylinder sides and tip. As shown in § 3.2, the generation of this vorticity requires consideration of the vorticity transport terms. Figures 6(b) and 6(e) show large magnitudes of $\overline{\omega _x}$ concentrated in regions downstream of the cylinder tip and base. These $\overline{\omega _x}$ concentrations extend downstream along the poles branching off the arch vortex. The present results show that the streamwise vorticity concentrations form the quadrupole and originate inside the arch vortex (i.e. significant concentrations exist for $x/D \leq 3$, within the vortex formation region).

Figure 6.

Time-averaged vorticity isosurfaces of (a) $\overline{\omega _y}D/U_{\infty }$, (b) $\overline{\omega _x}D/U_{\infty }$, (c) $\overline{\omega _z}D/U_{\infty }$ for $H/D = 4$ (top), and (d) $\overline{\omega _y}D/U_{\infty }$, (e) $\overline{\omega _x}D/U_{\infty }$, (f) $\overline{\omega _z}D/U_{\infty }$ for $H/D = 7$ (bottom), $\delta /D = 0.98$.

To clarify the origins of the vorticity distribution shown in figure 6 and the relationship between the instantaneous vortex dynamics and the mean field quadrupole structure, an analysis of the tilting and stretching terms in the vorticity transport equation was performed.

3.2. Vorticity transport analysis

For the flow of a viscous, incompressible fluid, the averaged vorticity transport equation is

(3.1) \begin{align} \overline{u_j} \frac{\partial \overline{\omega_i}}{\partial x_j} &= \underbrace{\overline{\omega_j} \frac{\partial \overline{u_i}}{\partial x_j}}_{\textit{mean stretching/tilting}} + \underbrace{\overline{\omega_j^{\prime} \frac{\partial u_i^{\prime}}{\partial x_j}}}_{\textit{fluctuating stretching/tilting}}\nonumber\\ &\quad + \frac{\partial}{\partial x_j} \Bigg(\underbrace{\nu \frac{\partial \overline{\omega_i}}{\partial x_j}}_{\textit{viscous diffusion}} - \underbrace{\overline{u_j^{\prime} \omega_i^{\prime}}}_{\textit{turbulence diffusion}}\Bigg). \end{align}

Table 2 summarizes the contribution of each term in (3.1) over the PIV domain. The quantities are shown as the fraction of the $\int _{V}({\textrm {D} \omega _i}/{\textrm {D} t} )\,\textrm {d} V$ over the PIV volume, $V$. In all directions, the viscous diffusion term is less than $2\,\%$, while the contributions due to the stretching/tilting terms are dominant. Of particular interest in this study are the terms responsible for the generation of $\overline{\omega _x}$, leading to the quadrupole structure observed in figures 5 and 6. The mean stretching/tilting terms are

(3.2)\begin{equation} \overline{\omega_j} \frac{\partial \bar{u}}{\partial x_j} = \underbrace{\overline{\omega_x} \frac{\partial \bar{u}}{\partial x}}_{\textit{mean stretching }\overline{S_x}} + \underbrace{\overline{\omega_y} \frac{\partial \bar{u}}{\partial y} + \overline{\omega_z} \frac{\partial \bar{u}}{\partial z}}_{\textit{mean tilting }\overline{T_x}}. \end{equation}

Table 2.

Volumetric average contribution of vorticity stretching, tilting and diffusion. Viscous diffusion was found to be less than $2\,\%$ in all directions.

The distributions of $\overline {T_x}$ and $\overline {S_x}$ are presented as isosurfaces in figure 7(a,b). High values of $\overline {T_x}$ and $\overline {S_x}$ are concentrated along the vortex cores corresponding to the quadrupole. The tilting terms act to increase the magnitude of $\overline{\omega _x}$ in the recirculating region, and the magnitude of $\overline {T_x}$ decreases rapidly downstream ($x/D > 4$). The stretching term $\overline {S_x}$ in figure 7(b) mainly amplifies $\overline{\omega _x}$ in the region $3 \leq x/D \leq 5$, which coincides with the end of the recirculation where ${\partial \bar {u}}/{\partial x}$ is largest. High values of $\overline {S_x}$ are concentrated along the poles.

Figure 7.

Time-averaged isosurfaces of (a) $\overline {T_x} D^{2}/U_{\infty }^{2}$, (b) $\overline {S_x}D^{2}/U_{\infty }^{2}$ and (c) $\overline {T_z}D^{2}/U_{\infty }^{2}$, for $H/D = 4$, $\delta /D = 0.98$.

Figure 7(c) shows the tilting term contribution to $\overline{\omega _z}$ (i.e. $\overline {T_z} = \overline{\omega _x} ({\partial \bar {w}}/{\partial x}) + \overline{\omega _y} ({\partial \bar {w}}/{\partial y})$); note that $\overline {S_z}= \overline{\omega _z} ({\partial \bar {w}}/{\partial z})$ is significantly smaller. The dominant contribution to $\overline{\omega _z}$ is due to reorientation of the vertical vorticity component $\overline{\omega _y}$ in the regions of the quadrupoles, resulting in an increase in the magnitude of $\overline{\omega _z}$ as seen in figure 6(c).

Based on mean field observations, it is possible to infer the influence of the tilting terms on the spatial evolution of the shed vortices. The process is illustrated with a simplified schematic in figure 8. The roll-up of the separated shear layer over the free end results in a region of predominantly $\overline{\omega _z}$ vorticity. Similarly, along the cylinder sides, the roll-up of the separated shear layer results in regions of predominantly $\overline{\omega _y}$ vorticity. Satisfying the solenoidal condition on the vorticity field, these high-vorticity regions are connected and observed as a continuous arch structure terminating at the ground. During the formation process, shown for $z>0$ in (8), the core of the forming vortex along the cylinder sides moves downstream. Along the arch legs of predominantly $\overline{\omega _y} > 0$ in figure 8(a), $\overline {T_x}>0$ in the base junction region, leading to the generation of $\omega _x > 0$; thus, the vorticity vector tilts downstream. Similarly, towards the free end, $\overline {T_x} < 0$, leading to the generation of $\omega _x < 0$, so that the vorticity vector tilts upstream.

Figure 8.

Schematic showing influence of tilting terms on mean vortex structure for $x/D \leq 4$ (only near side of wake is shown). (a) Perspective view showing influence of $\overline {T_x}$; (b) planar view showing influence of $\overline {T_z}$.

Near the free end, a small region of $\overline {T_x}>0$ reduces the magnitude of $\omega _x < 0$. From figure 8(b), in this region $\overline {T_z}<0$, so that the magnitude of $\omega _z<0$ increases and the vorticity vector is reoriented towards $z=0$, thereby satisfying the solenoidal condition. In the base region, $\overline {T_z}>0$ and the vorticity vector is slightly oriented towards $z=0$. The reorientation process occurs in the recirculating region, $x/D < 4$, before the structures are fully formed and shed.

The schematic illustration in figure 8 shows similarities to the model proposed by Wang & Zhou (Reference Wang and Zhou2009) for finite cantilevered square cylinders deduced from instantaneous PIV observations. That model implies that the upwash and downwash result in the bending of the spanwise (side) structures at the tip and base. Comparing figures 4 and 7(a), however, the present results indicate that the action of the tilting terms occurs upstream of the region of strong downwash/upwash. Moreover, it is observed that the magnitude of the tilting terms is insensitive to the downwash or upwash strength.

The mean field analysis and resulting schematic in figure 8 are representative of $4 \leq H/D \leq 7$. For $H/D = 3$, the mean field trends underlying figure 8 appear less clearly defined. Closer inspection of the temporal evolution of the wake shows that over most of the observation window, the vortex shedding process qualitatively matches that observed for $4 \leq H/D \leq 7$. However, there exist some short and irregularly occurring intervals over which the shedding process is characterized by much weaker vortices or is even suppressed. Consequently, the mean field imprint for $H/D = 3$ is still a quadrupole, but the instantaneous dynamics show strong modulations. This observed shedding modulation at $H/D=3$ is similar to that observed by Palau-Salvador et al. (Reference Palau-Salvador, Stoesser, Fröhlich, Kappler and Rodi2010) for thin turbulent boundary layers and higher Reynolds number, $\delta /D \approx 0.5, Re = 46\,000$, and $H/D=2.5$. Palau-Salvador et al. (Reference Palau-Salvador, Stoesser, Fröhlich, Kappler and Rodi2010) observed intervals of regular shedding interspersed with periods of weak and irregularly shed symmetric vortices or suppressed periodicity, suggesting that the present case $H/D = 3$ exhibits the inception of features related to lower-$H/D$ regimes.

3.3. Unsteady wake development

The time-resolved velocity field data were analysed using proper orthogonal decomposition (POD) in order to educe coherent structures from a low-dimensional representation of the flow. The method has been successfully applied for the analysis of tomographic PIV data (Morton, Yarusevych & Scarano Reference Morton, Yarusevych and Scarano2016; Zhu et al. Reference Zhu, Wang, Wang and Wang2017). The five most energetic modes were selected for the low-order representation of the flow. These modes consist of the first and second harmonic pairs as well as a slow-varying mode which is commonly observed in bluff body wakes (Hosseini, Martinuzzi & Noack Reference Hosseini, Martinuzzi and Noack2015). These modes account for 50 % to 75 % of the turbulent kinetic fluctuation energy (TKE) for the present cases. To evaluate the relative energy content in each 3D-3C POD mode for varying $H/D$, the total TKE was calculated for the first 100 modes. The results are plotted for $H/D = 3 - 7$ in figure 9. Studies which have examined the TKE distribution of the 2D square and circular cylinder have found that the first two POD modes make up a majority of the total TKE contribution (Van Oudheusden et al. Reference Van Oudheusden, Scarano, Van Hinsberg and Watt2005) and have been associated with vortex shedding. The present results show that as $H/D$ increases, the relative energy content of the first two modes increases. The results for $H/D = 5 - 7$ are similar to those for the 2D cylinder, with approximately 60 $\%$ of TKE captured in the first mode pair (Van Oudheusden et al. Reference Van Oudheusden, Scarano, Van Hinsberg and Watt2005). The results show that $H/D$ strongly influences the relative energy content of the first two modes. The present results suggest that for the range of $H/D$ and $\delta /D$ values studied herein, the full 3D shedding topology always exhibits anti-symmetric Von Kármán shedding; however, the energy contributions of the modes associated with these structures become gradually weaker with decreasing aspect ratio. For these cases, this results in a shift in energy content to higher mode numbers.

Figure 9.

Distribution of TKE in the first 100 POD modes for $H/D = 3 - 7$ and $\delta /D = 0.98$.

Figure 10(a) displays the POD temporal coefficients of modes 1 and 2. Modes 1 and 2 show strong periodic behaviour and are separated by a phase angle of ${\rm \pi} /2$, forming a mode pair. Analysis of the power spectral density (PSD) functions for these modes (figure 10b) shows a peak at the shedding frequency $f_{sh}$ corresponding to $\textit {St} = f_{sh}D/U_\infty = 0.16$, which is consistent with previous experimental studies for finite wall-mounted circular cylinders (Kuwamura et al. Reference Kuwamura, Hiwada, Hibino, Mabuchi and Kumada1984; Sumner et al. Reference Sumner, Heseltine and Dansereau2004).

Figure 10.

(a) POD temporal coefficients of modes 1 and 2 for $H/D = 4$ and $\delta /D = 0.98$. (b) PSD function of first and second temporal coefficients for modes 1 and 2. The quantity $t^{*} = t\cdot f_{sh}$ corresponds to the number of vortex shedding cycles.

Figure 11 shows isosurfaces of $Q=0$, coloured by $S_x$, and illustrates the topology of the shed structures for three time instances of a representative shedding cycle. The wake structure is found to be qualitatively similar for all cases studied herein. The figure shows that vortices are shed alternately from opposing sides of the cylinder. The ends of the shed vortex bend upstream. These ‘legs’ connect to the opposing vortex, forming a full loop, similar to the structures reported for cantilevered finite square cylinders (Bourgeois et al. Reference Bourgeois, Sattari and Martinuzzi2011; Hosseini et al. Reference Hosseini, Bourgeois and Martinuzzi2013). The full-loop structures appear to originate within the arch vortex, which grows streamwise as they are formed and eventually pinch off at $x/D \approx 4$. At this location, the poles are first identified emerging from the time-averaged arch vortex in figure 5.

Figure 11.

Flow structures are visualized with isosurfaces of $Q = 0$, coloured by $S_x$ in (a–f) and overlaid with mean field in (g–i). (a–c) Isometric view, (d–f) top view, (g–i) mean field (grey) and POD reconstruction (red) overlay. Results correspond to a finite cylinder wake for $H/D = 4$, $\delta /D = 0.98$, with (a) $t^{*} = 0$, (b) $t^{*} = 0.35$, (c) $t^{*} = 0.70$, (d) $t^{*} = 0$, (e) $t^{*} = 0.35$, (f) $t^{*} = 0.70$, (g) $t^{*} = 0$, (h) $t^{*} = 0.15$ and (i) $t^{*} = 0.30$.

Supplementary movie files show POD-based low-dimensional reconstructions of the shedding process. The $Q=0$ isosurfaces are coloured by $S_x$ (see supplementary movie 1, available at https://doi.org/10.1017/jfm.2021.904) and $T_x$ (see supplementary movie 2) to illustrate the spatial evolution of stretching and tilting. The stretching terms exhibit maxima at locations where structures pinch off from the arch vortex. Maxima in the tilting terms occur at locations where the legs of the full-loop structures bend back. These terms are largest for $x/D < 4$, and decrease rapidly in magnitude for $x/D > 4$. This behaviour is qualitatively shown in the reconstructed shedding sequence in figure 11, where much of the vortex reorientation is seen to occur for $x/D > 4$. These observations further support the view that the upwash, observed downstream of the arch vortex, is a result of the time-averaged signature of the passage of the vortex legs. The streamwise stretching term, $S_x$, is largest along the legs of the shed full loops in the region $4 < x/D < 5$. In this region, the spacing between subsequent vortices increases (i.e. the convective velocity has increased in the region as vortices are shed). However, the vortex legs must connect to subsequent vortices – so they become longer (i.e. are stretched). The time-averaged field and POD reconstruction are overlaid in figure 11, showing that the legs of the full-loop structures bend back along the poles of the time-averaged quadrupole (supplementary movie 3).