3.1 Thick cylinder with short splitter ( $AR=0.6$ and $c/(2a)=1$ )

This configuration is first characterized for low and high $I_{u}$ at various $Re$ and $AoA$ of the cylinder. Here, the splitter exhibits small-amplitude oscillations ( ${<}8^{\circ }$ , with the maximum achieved for the highest flow and turbulence levels) around the equilibrium position, which is nearly parallel to the direction of the mean flow. Spectral decomposition of the instantaneous angular velocity $\dot{\unicode[STIX]{x1D6FC}}$ of the splitter, $\unicode[STIX]{x1D6F7}_{\dot{\unicode[STIX]{x1D6FC}}}$ , reveals the dominant motions and the distinctive effects of the turbulence. This is illustrated in figure 2 with representative cases at $Re=3.8\times 10^{4}$ and cylinder with $AoA=30^{\circ }$ for the low and high $I_{u}$ . The $\unicode[STIX]{x1D6F7}_{\dot{\unicode[STIX]{x1D6FC}}}$ under low $I_{u}$ exhibits two distinctive peaks, namely $f_{p}$ and $f_{v}$ ; whereas $f_{v}$ is missing in the high-turbulence counterpart. This phenomenon is observed in all $AoA$ and $Re$ ; where the non-dimensional dominant pitching frequencies $f_{p}$ and $f_{v}$ are summarized in figure 3 as Strouhal number $St=f(2a)/U$ . In general, nearly constant $St$ is observed across the $Re$ , which reveals a linear increase of $f_{p}$ with $U$ regardless of $I_{u}$ . Also noted a linear relation for $f_{v}$ but at much larger growth rate. Jin et al. (Reference Jin, Ji, Liu and Chamorro2016b ) reported similar linear increase of pitching frequency with flat plates under uniform incoming flow. This suggests that the dynamics of the splitter can be approximated by the equation

where $\unicode[STIX]{x1D6FD}$ represents the damping of the system and $M_{w}$ embodies all the torques generated from the wind. Considering the modulation of the cylinder in the wake, we assume $M_{w}=M_{\unicode[STIX]{x1D6FC}\bar{u}}+M_{\unicode[STIX]{x1D6FC}u^{\prime }}$ , where $M_{\unicode[STIX]{x1D6FC}\bar{u}}$ and $M_{\unicode[STIX]{x1D6FC}u^{\prime }}$ indicate the angle-dependent moment generated from the mean and fluctuating velocities $\bar{u}$ and $u^{\prime }$ . For thin flat plates, $M_{\unicode[STIX]{x1D6FC}\bar{u}}=1/2C_{m}\unicode[STIX]{x1D70C}_{a}\bar{u}^{2}c^{2}L$ , where $C_{m}$ is the moment coefficient around the leading edge and $\unicode[STIX]{x1D70C}_{a}$ is the air density. Noting that the velocity distribution along the chord of the splitter can be non-uniform in the wake region, here $\bar{u}$ represents the effective velocity that embodies the averaged aerodynamic moment on the splitter. Considering a small-amplitude assumption, $C_{m}\approx -\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}/2$ , and ignoring other potential disturbances as a first-order approximation, the equation (3.1) can be rearranged as:

The equivalent counterpart in the Fourier domain, shown in (3.3), reveals insight on the dominating frequencies.

Indeed, note that when $f\rightarrow f_{p}=\sqrt{\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{a}L/I}(c\bar{u}/4\unicode[STIX]{x03C0})$ , the norm of the denominator is minimized and $\hat{\unicode[STIX]{x1D6FC}}(f_{p})$ is locally maximized; this also corresponds to the governing frequency observed in the pitching of a flat plate under uniform incoming flow (Jin et al. Reference Jin, Ji, Liu and Chamorro2016b ).

Under very low turbulence, the influence of vortex shedding from the elliptic cylinder results in periodic fluctuations of $M_{\unicode[STIX]{x1D6FC}u^{\prime }}$ , where $f_{v}$ dominates in ${\mathcal{F}}(M_{\unicode[STIX]{x1D6FC}u^{\prime }})$ . However, turbulence may severely dampen the coherence of the vortex shedding and lead to a broader spectral distribution (Tamura & Miyagi Reference Tamura and Miyagi1999) with $f_{v}$ non-distinctive, as shown in figure 2(b). Within the range of $Re$ investigated, both the vortex shedding frequency and $\bar{u}$ are proportional to the mean incoming flow. This leads to the quasi-linear increase of both $f_{p}$ and $f_{v}$ with $U$ , as reflected by the constant $St$ in figure 3. To further explore the influence of the splitter on the vortex shedding, $St$ of the bare elliptic cylinder under low $I_{u}$ is also included as a reference. This particular measurement shows good agreement with the experimental study by Modi et al. (Reference Modi, Wiland, Dikshit and Yokomizo1992), where $St$ is independent of $Re\in (3,10)\times 10^{4}$ . The comparison of $St$ with and without the splitter indicates that for the cylinder at $AoA=0^{\circ }$ , the small-amplitude pitching of the splitter results in a reduction of the vortex shedding rate $f_{v}$ of ${\approx}15\,\%$ , which is similar to a fixed splitter in the wake of a circular cylinder (Apelt & West Reference Apelt and West1975). However, for $AoA=30^{\circ },45^{\circ }$ and $60^{\circ }$ , the $f_{v}$ under the influence of the splitter is close (slightly higher) to the case with bare cylinder. Similar phenomenon was reported by Ozono (Reference Ozono1999); a distinctive rise of $St$ was observed with a short splitter attached to a circular cylinder and shifted half a diameter from the wake centreline. Roshko (Reference Roshko1955) pointed out that the vortex shedding frequency can be approximated as $f_{v}=St^{\ast }U/d^{\prime }$ , where $St^{\ast }$ is the wake Strouhal number and $d^{\prime }$ represents the distance between free streamlines in the wake region. As suggested by Roshko (Reference Roshko1955), $St^{\ast }$ remains nearly constant regardless of the cylinders geometry. Numerical and experimental studies with dual parallel splitters attached at the two transverse edges of circular and semicircular cylinders have shown that the flow separation point is shifted from the cylinder towards the trailing edge of the splitter. Here, the spanwise extent of the wake decreases which leads in a smaller $d^{\prime }$ compared to the bare cylinder case (Bao & Tao Reference Bao and Tao2013; Hamed et al. Reference Hamed, Vega, Liu and Chamorro2017). Further inspection suggests that $St^{\prime }=f(2a^{\prime })/U$ , where $2a^{\prime }$ is the frontal width of the elliptic cylinder under various $AoA$ , remains nearly constant for the short splitter; however, this does not hold for the bare case and that with the long splitter (shown later). This highlights the dominant influence of the spanwise extent of the wake on the vortex shedding frequency. Unlike the case of flat plate pitching under uniform flow, where the dynamics exhibits a quasi-sinusoidal pattern (Jin et al. Reference Jin, Ji, Liu and Chamorro2016b ), the $\dot{\unicode[STIX]{x1D6FC}}$ is very irregular even under low $I_{u}$ ; their standard deviation $\unicode[STIX]{x1D70E}_{\dot{\unicode[STIX]{x1D6FC}}}$ is provided in the subplots of figure 3(a–d). For all cases, $\unicode[STIX]{x1D70E}_{\dot{\unicode[STIX]{x1D6FC}}}$ increases monotonically with $Re$ due to higher velocity fluctuations. As expected, this quantity is larger with higher turbulence levels. Also, the differences of $\unicode[STIX]{x1D70E}_{\dot{\unicode[STIX]{x1D6FC}}}$ between the low and high $I_{u}$ increases with $Re$ under large cylinder $AoA$ ( $45^{\circ }$ and $60^{\circ }$ ; see subplots of figure 3). Note that the high asymmetry of the system at large $AoA$ and the increased plate pitching at higher $Re$ enhance the transverse flow and, consequently, wake mixing.

Figure 4.

2-D spectra of the streamwise velocity $\unicode[STIX]{x1D6F7}_{u}/[\unicode[STIX]{x1D6F7}_{u}]_{max}$ at various transverse locations downstream of the short splitter at $Re=3.8\times 10^{4}$ . (a) $AoA=0^{\circ }$ , low $I_{u}$ ;(b) $AoA=30^{\circ }$ , low $I_{u}$ ; (c) $30^{\circ }$ , high $I_{u}$ .

The structure of the velocity fluctuations in the wake of the cylinder–splitter system carries the signature of the energetic and dominant motions in the various configurations. This is illustrated in the 2-D spectra of the streamwise velocity fluctuations shown in figure 4 for selected cases. Under low $I_{u}$ , $f_{v}$ is dominant along the transverse direction; whereas the influence of the low-frequency pitching $f_{p}$ is reflected around $y/a=0$ for the cylinder at $AoA=0^{\circ }$ . Interestingly, at cylinder $AoA=30^{\circ }$ , the footprint of $f_{p}$ is presented at $y<0$ (figure 4 b). Numerical simulations of a splitter deflected to one side of a circular cylinder by Lu et al. (Reference Lu, Guo, Tang, Liu, Chen and Xie2016) indicates an asymmetric pressure distribution and transverse wake profile. The signature of $f_{p}$ induced by the splitter is then transmitted following the trace of the flow and shifted to $y<0$ . This ‘flow shifting’ is also noted for cylinder $AoA=45^{\circ }$ , and $60^{\circ }$ , not shown for brevity. Different signature arises under high $I_{u}$ (figure 4 c); there, $f_{v}$ is non-distinctive and the footprint of $f_{p}$ in the wake is not local, suggesting that the signal of slow-pitching model induced via $\bar{u}$ can be significantly disturbed by the surrounding turbulence.

3.2 Thick cylinder with long splitter ( $AR=0.6$ and $c/(2a)=2$ )

The increase of the splitter length to twice the semimajor axis of the cylinder leads to very distinctive motions; representative features are illustrated at $Re=3.8\times 10^{4}$ for the low $I_{u}$ in figure 5. Specifically, figure 5(a) shows $\unicode[STIX]{x1D6F7}_{\dot{\unicode[STIX]{x1D6FC}}}$ for the cylinder aligned with the flow, i.e. $AoA=0^{\circ }$ . The motion is characterized by a comparatively broad frequency distribution with no dominant peak. In this condition, the splitter undergoes chaotic-like behaviour with very small amplitude. Shukla et al. (Reference Shukla, Govardhan and Arakeri2009) reported that for splitters freely hinged in the centre of the wake of a circular cylinder, the periodic vortex shedding is suppressed with a significant decrease of pitching amplitude for splitter larger than ${\sim}3$ cylinder diameters. It is worth noting that the ratio of chord length to the hydraulic diameter of the elliptic cylinder $D_{H}=4A/P$ , where $A$ and $P$ are the cross-sectional area and perimeter of the ellipse, is close to 3. In contrast, the high background turbulence increases the flow mixing in the wake resulting in higher $\bar{u}$ ; this leads to an increased influence of $M_{\unicode[STIX]{x1D6FC}\bar{u}}\propto \bar{u}^{2}$ , sufficient for the $f_{p}$ signature. More energetic vortex shedding from the cylinder occurs at larger $AoA$ , and the dynamics of the splitter becomes highly influenced by $M_{\unicode[STIX]{x1D6FC}\bar{u}}$ and $M_{\unicode[STIX]{x1D6FC}u^{\prime }}$ . Figure 5(b) shows a representative spectrum of $\dot{\unicode[STIX]{x1D6FC}}$ for $AoA=45^{\circ }$ . In addition to $f_{p}$ , it shows two distinctive vortex shedding at frequencies $f_{v1}$ and $f_{v2}$ . We noted that similar phenomenon occurred with the cylinder $AoA=30^{\circ }$ and $60^{\circ }$ for all $Re$ . Knisely (Reference Knisely1990) experimentally investigated the body force fluctuations of a rectangular cylinder with aspect ratio $B/G=2$ , where $G$ and $B$ are the sides with the shorter one facing the flow. There, the fluctuating lift also exhibits multiple modes with two dominating peaks. Other studies with detailed flow visualizations including Laneville & Yong (Reference Laneville and Yong1983) and Yu et al. (Reference Yu, Butler, Kareem, Glimm and Sun2012) indicate that two vortex shedding modes coexist for the long splitter case. This is illustrated here in the subplot of figure 5(b); those modes correspond to a main vortex shedding starting close to the leading edge of the splitter from the cylinder ( $f_{v1}$ ) and a secondary vortex close to the trailing edge of the splitter ( $f_{v2}$ ). We further verified this with additional flow measurement in the vicinity of the leading edge of the splitter fixed parallel to the flow to exclude the influence of pitching. Figure 6 shows the lower vortex shedding frequency corresponding to the peak $f_{v1}$ close to the leading edge of the splitter; whereas the higher-frequency component $f_{v2}$ dominates in the wake region.

Figure 5.

Angular velocity spectra $\unicode[STIX]{x1D6F7}_{\dot{\unicode[STIX]{x1D6FC}}}$ of the long splitter at $Re=3.8\times 10^{4}$ and low $I_{u}$ with the cylinder at $AoA=$ (a) $0^{\circ }$ , (b) $45^{\circ }$ , and (c) $60^{\circ }$ ; (d) $AoA=60^{\circ }$ with splitter in inclined equilibrium.

Figure 6.

Velocity spectra $\unicode[STIX]{x1D6F7}_{u}$ close to the leading edge of the splitter (blue) and in wake region (black) at $Re=3.1\times 10^{4}$ , $AoA=60^{\circ }$ highlighting distinctive vortex shedding modes.

For all the cases discussed, the equilibrium position of the pitching is nearly coincident with the direction of the mean flow. Notably, there are two pitching equilibria for sufficiently high $AoA$ of the cylinder. Figure 5(c,d) illustrates the dynamics of the splitter with the cylinder at $AoA=60^{\circ }$ and low $I_{u}$ for the two equilibria. In the new splitter (inclined) equilibrium, the trailing edge of the splitter intersects with the downstream line from the leading edge of the elliptic cylinder; this is illustrated in the schematic of figure 5(d). The existence of multiple equilibria was first noted in hinged splitters in the wake of circular cylinders at sufficiently high $Re$ . In those scenarios, the stable equilibrium is in either side of the cylinder (Assi et al. Reference Assi, Bearman and Kitney2009; Gu et al. Reference Gu, Wang, Qiao and Huang2012; Lu et al. Reference Lu, Guo, Tang, Liu, Chen and Xie2016). Lācis et al. (Reference Lācis, Brosse, Ingremeau, Mazzino, Lundell, Kellay and Bagheri2014) pointed out that the recirculation zone behind the circular cylinder generates low pressure region at both sides of the wake, which allows a splitter to oscillate in either side. In our measurements, the larger and stronger recirculation at large cylinder $AoA$ , e.g. ${\sim}45^{\circ }$ or larger (Paul, Arul Prakash & Vengadesan Reference Paul, Arul Prakash and Vengadesan2014), allows the inclined equilibrium. In this inclined state, the dual vortex shedding mode vanishes and only one distinctive vortex shedding frequency component is prominent. To further explore the effect of the secondary equilibrium, complementary measurement of the lift and drag forces were carried out on the cylinder. Lācis et al. (Reference Lācis, Brosse, Ingremeau, Mazzino, Lundell, Kellay and Bagheri2014) pointed out that a drift force towards the inclined splitter is produced as a result of the symmetry breaking. Smoke flow visualizations by Gu et al. (Reference Gu, Wang, Qiao and Huang2012) suggested that a lower pressure region compared to the other equilibrium counterpart is generated between the base of the cylinder and the splitter. As illustrated in the schematics of figure 7, this lower pressure region results in stronger drag $D$ and lift $L$ forces in $y<0$ direction as reflected by the averaged drag and lift coefficients $C_{d}=D/(\unicode[STIX]{x1D70C}_{a}U^{2}Ha)$ and $C_{l}=L/(\unicode[STIX]{x1D70C}_{a}U^{2}Ha)$ across $Re$ ; here, $H$ denotes the span of the cylinder. The cylinder–splitter system can stay in either equilibrium depending on the initial location of the splitter or the external disturbances. The distinctive variation of the $C_{d}$ and $C_{l}$ between the two equilibria may induce instability on the structure when hinged splitters are used as a passive flow control mechanism.

Figure 7.

Drag $D_{d}$ and lift $C_{l}$ coefficients at $AoA=60^{\circ }$ with the splitter equilibrium (a) parallel to the flow and (b) inclined towards the cylinder leading edge. The schematics highlight the low pressure region in each case.

Figure 8.

Dimensionless pitching frequencies $f_{p}$ , $f_{v1}$ and $f_{v2}$ of the long splitter with cylinder at $AoA=$ (a) $0^{\circ }$ , (b) $30^{\circ }$ , (c) $45^{\circ }$ , (d) $60^{\circ }$ . The subplots in (c) and (d) show the inclined cases under low $I_{u}$ .

The normalized $f_{p}$ and $f_{v}$ as a function of the $Re$ is summarized in figure 8. Similar to the short splitter, the evolution of $f_{p}$ , $f_{v1}$ and $f_{v2}$ exhibit a quasi-linear relation with $U$ (figure 8 a–d). It is possible to infer that the ratio $f_{v2}/f_{v1}$ decreases from 1.92 to 1.64 when the $AoA$ of the cylinder increases from $30^{\circ }$ to $60^{\circ }$ , suggesting a converging trend to $f_{v2}/f_{v1}=1.625$ measured by Knisely (Reference Knisely1990) in rectangular plates with $B/G=2$ ; this roughly resembles the case of the $AoA=90^{\circ }$ . Similar to the short splitter, i.e. $c/(2a)=1$ , the influence of periodic vortex shedding from the cylinder is non-distinctive for all the high $I_{u}$ cases. Interestingly, $f_{p}(Re)$ with the long splitter exhibits larger differences between high and low $I_{u}$ cases compared to those with the short splitter. This suggests a stronger wake mixing with incoming turbulence and, therefore, higher $\bar{u}$ compared to the low $I_{u}$ counterpart at $x/(2a)\approx 2$ . Similar flow recovery with turbulence has been reported in various cases including wind turbine wakes (Jin et al. Reference Jin, Liu, Aggarwal, Singh and Chamorro2016c ) and rectangular cylinders (Naimul Haque et al. Reference Naimul Haque, Katsuchi, Yamada and Nishio2014). The $f_{p}$ and $f_{v}$ for the inclined equilibrium as a function of $Re$ under low $I_{u}$ is given in the subplots of figure 8(c,d). Here, $f_{p}$ remains almost the same between the two equilibria, suggesting that the effective averaged velocity $\bar{u}$ is not sensitive to the orientation of the splitter. It is noted that the high $I_{u}$ significantly reduces the recirculation region and induces flow disturbances (Naimul Haque et al. Reference Naimul Haque, Katsuchi, Yamada and Nishio2014; Liu et al. Reference Liu, Hamed, Jin and Chamorro2017), which hinders the formation of an inclined equilibrium; the inclined state was also observed at $AoA=45^{\circ }$ and $60^{\circ }$ for the short splitter. However, due to the geometry of the elliptic cylinder, the large inclination of the splitter resulted in a collision with the cylinder, affecting the unsteady pitching. The wake signature at various transverse locations within $y/a\in [-1,1]$ is highlighted in the 2-D spectra of the streamwise velocity fluctuations $\unicode[STIX]{x1D6F7}_{u}$ for selected cases in figure 9. At cylinder $AoA=0^{\circ }$ (figure 9 a) and low $I_{u}$ , there is no clear dominant frequency across the wake region. For cases where the splitter motion contain dominant $f_{v1}$ and $f_{v2}$ , it is interesting to note that although $f_{v2}$ is less energetic in the splitter pitching (figure 5 b,c), it is prominent in the wake. Knisely (Reference Knisely1990) also reported a reversed order of the predominant peak from fluctuating lift and wake in a rectangular plate of aspect ratio $B/G=2$ . This, with flow visualizations of Laneville & Yong (Reference Laneville and Yong1983) and Yu et al. (Reference Yu, Butler, Kareem, Glimm and Sun2012), suggests that the primary vortex shedding from the elliptic cylinder may be severely disturbed before reaching the wake due to a reattachment with the splitter. Also, secondary vortex generated near the splitter tip ( $f_{v2}$ ) may increase wake fluctuation. The wake spectra of the corresponding inclined splitter is presented in figure 9(c); there, the influence of $f_{p}$ appears at $y>0$ coincident with the ‘flow shifting’ observed at the parallel equilibrium of the splitter.

Figure 9.

2-D spectra of the streamwise velocity $\unicode[STIX]{x1D6F7}_{u}/[\unicode[STIX]{x1D6F7}_{u}]_{max}$ at various transverse locations downstream of the long splitter at $Re=3.8\times 10^{4}$ under low $I_{u}$ . Cylinder at $AoA=$ (a) $0^{\circ }$ , (b) $45^{\circ }$ , (c) $45^{\circ }$ with splitter in inclined equilibrium.

3.3 Thin cylinder $(AR=0.3)$

As discussed above, sufficient recirculation in the wake can result in an inclined equilibrium of the splitter; complementary measurements with a slender elliptic cylinder of $AR=0.3$ stress the phenomenon. Figure 10(a) and subplot show the existence of a second equilibrium at $AoA=30^{\circ }$ and low $I_{u}$ for the short splitter, where no inclination occurred for the thick cylinder. It is noted that elliptic cylinders with low $AR$ facilitate more abrupt flow separation at the leading edge, which induce longer and stronger recirculation regions (Paul et al. Reference Paul, Arul Prakash and Vengadesan2014). The extended recirculation can result in splitter inclination even under high background turbulence. Indeed, as illustrated in figure 10(b) for $AoA=60^{\circ }$ , the splitter pitches stably when inclined to the leading edge side under high $I_{u}$ . Note that the more severe flow separation results in stronger vortex shedding from cylinder, where the corresponding frequency component $f_{v}$ in the dynamics of the splitter is noticeable even under high $I_{u}$ .