4.1. Aeroacoustic and aerodynamic characteristics

Far-field noise PSD, $\varPhi _{pp}$ (dB Hz⁻¹), derived from the experimental noise data according to (2.3), is presented in figure 5( $a$ ), which demonstrates a dominant peak occurring at $St = 0.142$ for the straight corner-oriented hexagonal cylinder (the baseline case $\gamma ^* = 0$ ). This corresponds to the fundamental Kármán vortex shedding from the straight cylinder as defined in (3.7b ), hereafter denoted as $St_0$ , which agrees well with previous LES (Masoudi et al. Reference Masoudi, Gan and Sims-Williams2021) and experimental (Xu et al. Reference Xu, Zhang, Gan, Li and Zhou2017) studies. When twisting is introduced, all $\gamma ^* \gt 0$ cases are shown to reduce both the tonal and broadband noise levels relative to the baseline, where the case $\gamma ^* = 1/6$ demonstrates a remarkable tonal noise reduction of up to 15 dB. A noteworthy observation is that for the cases displaying a single dominant peak, $\varPhi _{pp}$ is not a monotonic function of $\gamma ^*$ , i.e. $\varPhi _{pp}|_{\gamma ^*=0}\gt \varPhi _{pp}|_{\gamma ^*=1/3}\gt \varPhi _{pp}|_{\gamma ^*=1/6}$ .

Figure 5.

( $a$ ) Far-field noise PSD, $\varPhi _{pp}$ , for varying $\gamma ^*$ cases. ( $b$ ) Dependence of the number of tones on $\gamma ^*$ , where the marker size is proportional to the peak $\varPhi _{pp}$ value. Experimental data at $ \textit{Re} = 2\times 10^4$ .

Another interesting observation is the change of the number of tones in the noise spectrum as $\gamma ^*$ increases. The tone number increases from one for the baseline case ( $\gamma ^*=0$ ) to two as $\gamma ^*$ reaches $1/18$ , and then remains at two until $\gamma ^*\lesssim 1/6$ , beyond which the spectrum reduces back to a single tone at $St_0$ , larger than that of the baseline yet with lower $\varPhi _{pp}$ . The dual tone observed within $1/18\leqslant \gamma ^*\lesssim 1/6$ suggests a vortex shedding pattern deviating from the classical Kármán type, which is plausibly attributed to spatial variation of spanwise twist, rather than to a temporal switch of vortex-shedding modes. These two patterns are labelled as Patterns I and II, respectively, as shown in figure 5( $b$ ), which will be elaborated in § 4.3. The frequencies for the two tones in this $\gamma ^*$ range are denoted as $St_1$ and $St_2$ , respectively. Exactly at which $\gamma ^*$ the second tone starts to emerge in the range $0\lt \gamma ^*\lt 1/18$ cannot be convincingly determined from the experiments due to the spatial constraint of the wind tunnel as explained in § 2. A close inspection of the inset plot in figure 5( $a$ ) reveals that the spectrum for the case $\gamma ^* = 1/6$ appears to lie between the single-tone and dual-tone cases. It can be interpreted either as a single, blunt hump or as two peaks without a clearly defined valley between them. This feature, as will be shown later, leads to a promising simultaneous noise reduction effect (aeroacoustically) and a stabilising effect (aerodynamically), with the optimal performance apparently occurring near $\gamma ^* = 1/6$ .

Twisted cylinders can be viewed as cylinders whose given cross-sectional shape (a hexagon in this case) rotates in the same direction along the span with an infinitesimal step, resulting in a continuously changing sectional incidence $\phi (z)$ and, in turn, a varying sectional flow separation point. Masoudi et al. (Reference Masoudi, Gan and Sims-Williams2021) found that for the straight hexagonal cylinder the vortex shedding frequency varies smoothly in the range $0.14\lesssim St_0 \lesssim 0.17$ as the incidence angle changes. When twist is applied, $St_1$ and $St_2$ in Pattern II likely originate from a similar range of incidence angles. In this regime, the corresponding frequencies exhibit a bimodal distribution centred around two preferred values of $St_1 \approx 0.14$ and $St_2 \approx 0.17$ , rather than a broadband spectrum. As the degree of twist further increases to $\gamma ^*\geqslant 1/5.4$ , only one preferred frequency is selected, which is $St_0$ in Pattern I, with its value ( $\approx 0.16$ ) almost equivalent to the arithmetic mean of $St_0$ ( $\approx 0.157$ ) across all incidence angles of a straight hexagonal cylinder.

The total far-field noise was evaluated by integrating the PSD $\varPhi _{pp}$ over $ 0.1\lesssim St\lesssim 5$ , yielding the OASPL. The effect of spanwise twist on noise reduction with respect to the baseline case ( $\Delta$ OASPL) is obvious, as shown in figure 6( $a$ ). The nonlinearly varying $\Delta$ OASPL with $\gamma ^*$ minimises at $\gamma ^* = 1/6$ , achieving a total noise reduction increasing with $ \textit{Re}$ , viz., 11 dB at $ \textit{Re}=2\times 10^4$ and 17 dB at $ \textit{Re}=10^5$ . In addition, the dependence of OASPL on the cylinder orientation, measured by the midspan incidence $\theta _0$ , is much weaker than on the twist angle, which is demonstrated in figure 7. As shown in figure 6( $b$ ), the far-field acoustic radiation exhibits a consistent scaling with the $Ma^6$ law regardless of $\gamma ^*$ , a characteristic of dipole sound sources, suggesting that the unsteady forces on the cylinder surface are the primary noise source. Within the tested subcritical $ \textit{Re}$ range, the case with $\gamma ^* = 1/6$ consistently shows the best noise reduction performance, underscoring the potential practical value of this specific twist angle.

Figure 6.

( $a$ ) Dependence of measured overall sound pressure level (OASPL) reduction on $\gamma ^*$ at $ \textit{Re} = 2\times 10^4$ , $6\times 10^4$ and $1\times 10^5$ . ( $b$ ) Variation of OASPL with Reynolds number for varying $\gamma ^*$ cases.

Figure 7.

Variation of far-field OASPL with cylinder midspan incidence angle $\theta _0$ and $\gamma ^*$ . Experimental data at $ \textit{Re} = 2\times 10^4$ .

Figure 8.

Comparison of aerodynamic performance for the twisted and straight cylinders at $ \textit{Re} = 2\times 10^4$ , obtained from numerical simulations. ( $a$ , $b$ ) Time history of drag and lift coefficients, ( $c$ ) phase-space plot of the force coefficients, ( $d$ ) PSD of lift coefficient fluctuation.

The far-field OASPL is associated with the flow field whose global characteristics can be captured by the time history of $C_{\kern-1pt D}$ and $C_{\kern-1pt L}$ , as shown in figure 8( $a$ , $b$ ), where $t^*=tU_\infty /D_i$ . These simulation results suggest that their fluctuation level, e.g. the root-mean-square (r.m.s.) values calculated based on 30 shedding cycles, namely $C'_{L,\textit{rms}}$ and $C'_{D,\textit{rms}}$ , as well as the time-averaged drag, $\overline {C}_{\kern-1pt D}$ , are significantly reduced for twisted cylinders, perhaps as expected. In line with the noise spectrum is the non-monotonic variation of the fluid forces with increasing $\gamma ^*$ . To facilitate a clear comparison of their features in time domain, figure 8( $c$ ) shows the phase diagram, where the strongest suppression of the oscillation amplitude of the two fluid forces for the $\gamma ^* = 1/6$ case is evident. In particular, $C'_{L,\textit{rms}}$ , $C'_{D,\textit{rms}}$ and $\overline {C}_{\kern-1pt D}$ are reduced by 90 %, 27 % and 37 %, respectively, compared with the baseline case. The $\overline {C}_{\kern-1pt D}$ value is also lower than that of the straight hexagonal cylinder in any orientation, which is typically $\gtrsim 1.5$ under a similar subcritical $ \textit{Re}$ (Masoudi et al. Reference Masoudi, Gan and Sims-Williams2021).

The PSD of $C_{\kern-1pt L}'$ for the four cases are shown in figure 8( $d$ ), with the $\gamma ^* = 1/6$ case exhibiting the lowest dominant spectral peak. This supports the finding of the most significant far-field noise attenuation for this case, as shown in figure 6( $a$ ), since the dominant acoustic radiation perpendicular to the free stream is associated with the fundamental frequency of lift fluctuations in a dipole source (Gerrard Reference Gerrard1955; Etkin, Korbacher & Keefe Reference Etkin, Korbacher and Keefe1957). Consistent with the behaviour of $\varPhi _{pp}$ , the case with $\gamma ^* = 1/9$ displays two distinct peaks, reinforcing the link between the near-field flow dynamics and far-field noise.

The aerodynamic and aeroacoustic results presented in this section therefore suggest that the twist angle $\gamma ^* = 1/6$ somehow yields the optimal control effect on a hexagonal cylinder, which minimises the tonal peak, OASPL, $\overline {C}_{\kern-1pt D}$ and r.m.s. forces at the same time, among all tested twist angles including the straight cylinder (zero twist), in all possible orientations. This conclusion plausibly applies up to $ \textit{Re}\sim 10^5$ , according to the trend of OASPL shown in figure 6( $a$ ). The physical mechanism behind this observation will be elucidated subsequently.

4.2. Shear layer separation along the span

The separation–reattachment behaviour of the shear layer can be determined by examining wall shear stress $\tau _{w} = \mu ({\mathrm{d}u_\Vert }/{\mathrm{d}y_\perp } )$ , where $u_\Vert$ is the local velocity component parallel to the wall and $y_\perp$ is the normal distance to the local wall. Figure 9 presents the distribution of the normalised time-averaged wall shear stress $\tau _{w}^* = \overline {\tau }_{w} / (\rho _0 f_{0}^2 D_i^2 )$ on the cylinder surface, with respect to the azimuthal angle $\theta$ defined in figure 1( $c$ ), where the characteristic $f_0$ represents the dominant vortex shedding frequency of the baseline straight cylinder. Marked lines that feature the flow separation characteristics, viz., the primary separation line (PSL) and secondary separation line (SSL), are determined by the magnitude ( $\overline {\tau }_{w}=0$ ) and the change of sign of $\overline {\tau }_{w}$ (Wu, Ma & Zhou Reference Wu, Ma and Zhou2015), respectively.

Figure 9.

Contours of normalised wall shear stress on the cylinder surface: ( $a$ ) $\gamma ^* = 0$ , ( $b$ ) $1/9$ , ( $c$ ) $1/6$ , ( $d$ ) $1/3$ . Solid lines mark the stagnation line along the span (SL); similarly, thick dashed lines mark the PSL, thin dashed lines mark the SSL and dotted lines are for the reattachment line (RL), all based on time-averaged flow fields. The azimuthal angle $\theta$ is defined in figure 1( $c$ ).

Figure 10.

Contours of $\overline {\omega }_z^*$ for the case $\gamma ^*=1/3$ in (i) and $\gamma ^*=1/6$ in (ii). ( $a$ ) Sections where shear layer separates without reattachment, corresponding to RI in figure 9; ( $b$ ) sections where shear layer reattaches and separates for the second time, corresponding to RII in figure 9. The windward stagnation points are labelled with solid circles. The sectional PSL, SSL and RL are labelled as primary separation points (PSP) with open circles, secondary separation points (SSP) with open squares and reattachment points (RP) with open pentagrams, respectively. The red dot in ( $a$ ) marks the ( $x,y$ ) location of the spanwise sampling probes for figure 13( $a$ i– $a$ iv). Here $\phi ^*(z) = \phi (z)/\phi _0$ .

According to the $\overline {\tau }_{w}$ distribution along the $z$ -direction, showing an antisymmetric pattern about $z/L_s=0.5$ , the cylinder midspan, two distinct separation regimes can be identified, which are labelled in figure 9( $b$ ) for example. In Regime I (RI), flow separates at the upper and lower PSL without reattachment, while Regime II (RII) features the flow reattachment downstream of PSL and separation for the second time at SSL. The spanwise range of RI, centred at the midspan, is determined by the appearance of SSL and is nonlinearly $\gamma ^*$ dependent, which are $0.81L_s$ , $0.84L_s$ and $0.71L_s$ , respectively, for $\gamma ^*=1/9$ , $1/6$ and $1/3$ , maximising at the case $\gamma ^*=1/6$ . This case also reveals the lowest $\overline {\tau }_{w}$ magnitude towards the base area, i.e. $\theta \to \pm 180^{\circ }$ , indicating a reduced influence from the vortices shed into the wake. This suggests that either the strength of these vortices has weakened or their roll-up occurs farther downstream.

The time-averaged classical Kármán vortex shedding, characterised by the shear layer separation–reattachment mechanism, is visualised in the $x$ – $y$ plane in figure 10, based on the normalised time-mean spanwise vorticity $\overline {\omega }_z^* = \overline {\omega }_z D_i / U_{\infty }$ , where $\omega _z= ({\partial v}/{\partial x} - \partial u/{\partial y} )$ . Recalling the antisymmetric nature of the sectional incidence angle $\phi (z)$ and the time-mean shear layer shape about $z/L_s=0.5$ across all cases, figure 10 reveals that at the same sectional incidence (i.e. within the same column), shear layers appear thinner and longer for the case $\gamma ^*=1/6$ compared with $\gamma ^*=1/3$ .

Masoudi et al. (Reference Masoudi, Sims-Williams and Gan2023) found that the length of the time-averaged shear layer, denoted as $\overline {L}_\omega$ and shown in figure 10( $b$ ), is an important length scale to characterise both the wake behind polygonal cylinders and the associated fluid forces. It can be quantified as the distance from the cylinder centre to the shear layer’s endpoint, based on a properly chosen threshold, which should not impact the finding sensitively. In the present study $\overline {\omega }^*=\pm 10$ , and the spanwise variation of $\overline {L}_\omega$ for the top and bottom shear layers are presented in figure 11( $a$ , $b$ ), respectively, which are antisymmetric to each other. The qualitative comparison of $\overline {L}_\omega$ visualised in figure 10 is now reassured quantitatively. That is, for most of the cylinder span, and hence local twist $\phi (z)$ , $\overline {L}_\omega$ of the shear layer behind the cylinder of $\gamma ^*=1/6$ is approximately $30\,\%$ larger than that of $\gamma ^*=1/3$ . In fact, it is evident that the former has the largest $\overline {L}_\omega$ along its span among all cases, and as long as twist is introduced (i.e. $\gamma ^*\gt 0$ ), $\overline {L}_\omega$ enlarges in general. By sorting the magnitudes, it can be seen that $\overline {L}_\omega |_{\gamma ^*=1/6}\gt \overline {L}_\omega |_{1/9}\gt \overline {L}_\omega |_{1/3}\gt \overline {L}_\omega |_{\gamma ^*=0}$ , following a nonlinear order that is opposite to that of $\overline {C}_{\kern-1pt D}$ shown in figure 8( $a$ ), consistent with the general relationship between $\overline {L}_\omega$ and $\overline {C}_{\kern-1pt D}$ reported in Masoudi et al. (Reference Masoudi, Gan and Sims-Williams2021). According to Gerrard (Reference Gerrard1966), an increase in turbulence intensity within the shear layer enhances diffusion in their interaction region, which reduces entrainment into the shear layer and shifts the vortex roll-up downstream, resulting in a longer shear layer. Consequently, the largest $\overline {L}_\omega$ in the case $\gamma ^* = 1/6$ suggests that the entrainment effect is weaker than in other cases, indicating a longer vortex formation length with weaker oscillations in the transverse direction.

Figure 11.

Time-averaged shear layer length $\overline {L}_\omega$ along the span for the top ( $a$ ) and bottom ( $b$ ) shear layers, whose spanwise distributions appear approximately antisymmetric. The red solid pentagram in each case marks the location where shear layer reattachment occurs.

Focusing on the twisted cylinder cases shown in figure 11( $a$ ), as the local twist angle $\phi$ increases along the $z$ -axis, $\overline {L}_\omega$ notably decreases until the spanwise location marked by the red solid pentagram, where shear layer reattachment is just about to occur and $\overline {L}_\omega$ reaches the minima. This range corresponds to RI illustrated in figure 9. Beyond the pentagram enters RII, where shear layer reattachment occurs and $\overline {L}_\omega$ experiences abrupt increment. After reattachment, the shear layer length increases sharply with a steep slope. This suggests that compared with RI, the entrainment effect in RII is weaker due to reattachment, leading to more stable shear layers. It may also be inferred from the rather invariant shear layer thickness observed for different $\gamma ^*$ and $\phi ^*$ cases that strong shear layer flapping motion, typically present in straight cylinders at some orientations (Masoudi et al. Reference Masoudi, Sims-Williams and Gan2023), is largely suppressed when twist is introduced. The flapping motion would otherwise lead to a thicker shear layer in a time-averaged sense.

Figure 12.

Contours of $v'_{\textit{rms}}$ in the $y=0$ plane overlaid with the spanwise variation of $L_f$ : ( $a$ ) $\gamma ^* = 0$ , ( $b$ ) $\gamma ^* = 1/9$ , ( $c$ ) $\gamma ^* = 1/6$ , ( $d$ ) $\gamma ^* = 1/3$ . In the plane shown, the distribution of $v'_{\textit{rms}}$ is similar to that of the total TKE, due to the dominance of the former. The RI/RII partition follows that in figure 9.

The spanwise variation in shear layer separation points can lead to local phase differences in vortex shedding, resulting in highly 3-D wake flow structures. These structures are reflected in the distribution of the r.m.s. transverse velocity fluctuation, $v'_{\textit{rms}}$ in the $y=0$ plane (Bevilaqua Reference Bevilaqua1973), which is shown in figure 12. Spanwise variation of the vortex formation length, $L_f(z)$ , defined as the $x$ -coordinate of the maximum $v'_{\textit{rms}}$ in this plane, is also marked on top of the $v'_{\textit{rms}}$ contour, going along the peak values. The distribution of $v'_{\textit{rms}}$ in figure 12( $a$ ) shows that the baseline case ( $\gamma ^* = 0$ ) exhibits a nearly spanwise invariant vortex shedding pattern. In contrast, variation of $v'_{\textit{rms}}$ along the span is evident for the twisted cases (figure 12 $b$ – $d$ ). The magnitude of maximum $v'_{\textit{rms}}$ follows a descending order: $\gamma ^*=0$ , $1/3$ , $1/9$ , minimising at $\gamma ^*=1/6$ , in line with the ordering of $C_{\kern-1pt L}'$ in figure 8( $b$ ) and $\Delta$ OASPL in figure 6( $a$ ), as expected from the demonstrated correlation between $v'_{\textit{rms}}$ and vortex shedding noise (Maryami & Liu Reference Maryami and Liu2024). The spanwise-averaged $L_f$ also follows the same order, but in reverse (ascending) direction. These trends indicate that the twisted cylinders attenuate turbulent fluctuations in the wake while pushing the wake region with strong TKE farther downstream.

In addition to the Kármán vortex shedding in the wake, a Kelvin–Helmholtz (K–H) type of instability emerges in the shear layer region. This instability originates from the velocity inflection point within the shear layer and grows as the shear layer rolls up to form the primary Kármán vortex street. The broadband peak observed in the velocity spectrum is attributed to this convective instability (Prasad & Williamson Reference Prasad and Williamson1997). To identify this phenomenon, velocity probes were deployed along the span of the cylinders, i.e. $(x,y,z)=(0.35D_i,0.85D_i,[0,L_s])$ as shown in figure 10( $a$ ). This line ensures that the probes are approximately within the shear flow region for all $\gamma ^*$ cases. The PSD of the $v'$ signal, $\varPhi _{\mathit{vv}}$ in decibels per hertz, was then obtained using Welch’s method with a reference velocity of 1 $\mathrm{m\,s}^{-1}$ and a resolution of 4 Hz.

Figure 13.

Contours of $v'$ PSD, $\varPhi _{\mathit{vv}}$ , along the spanwise direction: (a i–a iv) near the shear layer region $(0.35D_i,0.8D_i,[0,L_s])$ , marked by the red dot in figure 10( $a$ ), and (b i–b iv) in the wake region $(3D_i,0,[0,L_s])$ . Labels i–iv represent $\gamma ^* = 0$ , $\gamma ^* = 1/9$ , $\gamma ^* = 1/6$ and $\gamma ^* = 1/3$ , respectively. The RI/RII partition follows that in figure 9.

Figure 13( $a$ i– $a$ iv) illustrates the spanwise dependence of $\varPhi _{vv}$ from the shear layer probes. The formation of K–H vortices is reflected by the spectral hump of $\varPhi _{vv}$ in the region $St \gtrsim 1$ (Prasad & Williamson Reference Prasad and Williamson1997; Lander et al. Reference Lander, Moore, Letchford and Amitay2018; Rajagopalan & Antonia Reference Rajagopalan and Antonia2005), for $\gamma ^*\gt 0$ cases (figure 13 $a$ ii– $a$ iv). Discernibly, the K–H instability is pronounced in RI (cf. figure 9) where statistically the separated shear layer is less likely to reattach, whereas shear layer reattachment (RII) tends to suppress the K–H instability. This lends support to the observation that flow reattachment stabilises the shear layer, consistent with the increased $\overline {L}_\omega$ shown in figure 11. For the straight cylinder, figure 13 ( $a$ i), the spectral distribution of $\varPhi _{vv}$ is much more homogeneous than the other cases for $St\gt St_0$ , despite at moderate levels. This is because the K–H instability in the straight cylinder is relatively weaker than the other cases, and is overshadowed by higher-frequency signals.

For comparison, $\varPhi _{vv}$ in the central wake region is illustrated in figure 13( $b$ i– $b$ iv), where the probes were positioned in the wake along the span, i.e. $(3D_i,0,[0,L_s])$ . In the baseline case $\gamma ^* = 0$ (figure 13 $b$ i), a strong peak appears constantly along the span at $St_0$ . As twist applies, two distinct vortex shedding frequencies $St_1$ and $St_2$ can be identified in $\gamma ^* = 1/9$ ( figure 13 $b$ ii), in RI and RII, respectively. In ( figure 13 $b$ iii), the increased twist of $\gamma ^* = 1/6$ results in a diminished $St_0$ peak in RI, especially near $z/L_s=0.5$ , consistent with the relatively broad, weak peak in the shear layer region ( figure 13 $a$ iii), and the maintained (strong) PSD level at $St_0$ in RII. Interestingly, as twist further increases to $\gamma ^*=1/3$ , shear layer attachment features (RI and RII region) lose their impact on the $\varPhi _{vv}$ distribution, which appears similar to that of the baseline case.

These observations plausibly explain the origin of tonal characteristics observed in figure 5. That is, Pattern I with a single dominant tone, represented by cases $\gamma ^*=0$ and $1/3$ , is attributed to the classical Kármán vortex shedding across the entire cylinder span, while $St_1$ in Pattern II ( $\gamma ^*=1/9$ ) is contributed by the primary flow separation (without reattachment) in RI, and the higher $St_2$ originates from the shear layer reattachment behaviour occurred in RII. The blunt peak in the $\gamma ^*=1/6$ case primarily results from the broader yet weaker $\varPhi _{vv}$ peak in RI. These characteristics are all due to the spatial effect because of spanwise twist. A careful inspection of figure 13( $a$ ii, $b$ ii) reveals that the two peaks in fact coexist in RII for $\gamma ^*=1/9$ , with $St_1$ weaker. This is a reflection of a temporal effect, confirmed by a local wavelet analysis (figure not shown), indicating that the shear layer reattachment behaviour in RII is intermittent in nature. More direct evidence will be provided in § 4.5.

4.3. Vortex structures in the $w$ ake

Turbulent wake flow structures behind the four cylinders are displayed in figure 14, where the vortex isosurface is based on the $Q$ -criterion. To facilitate direct comparison, the flow field is replicated on both ends in cases $\gamma ^*=0,1/6$ and $1/3$ as per the periodic boundary condition, so that all four cases appear to have the same displayed span as the $\gamma ^*=1/9$ case. Behind the straight cylinder in (figure 14 $a$ ), classical Kármán vortex shedding is evident, with vortices appearing more organised as they evolve downstream. Ribbon-like streamwise vortices are noticeable, which were also present in straight cylinders of other cross-sectional geometries, e.g. the heptagonal (Masoudi et al. Reference Masoudi, Sims-Williams and Gan2023) and circular (Aljure et al. Reference Aljure, Lehmkhul, Rodríguez and Oliva2017) cylinders at similar Reynolds numbers.

Figure 14.

Isosurface of normalised $Q$ -criterion, $Q_n = QD_i^2/U_{\infty }^2 \approx 0.09$ , coloured by normalised spanwise vorticity, $\omega _z D_i/U_\infty$ : ( $a$ ) $\gamma ^* = 0$ , ( $b$ ) $\gamma ^* = 1/9$ , ( $c$ ) $\gamma ^* = 1/6$ , ( $d$ ) $\gamma ^* = 1/3$ . For direct comparison, cylinder spans in ( $a$ ), ( $c$ ) and ( $d$ ) have been replicated symmetrically at both ends in accordance with the periodic boundary condition, so as to match the simulated span of the $\gamma ^*=1/9$ case in ( $b$ ), which has the largest twist wavelength $L_s$ (see § 3.2).

The nonlinear effect of the twist angle $\gamma ^*$ is also clearly reflected in the wake flow structures. Introducing a weak twist of $\gamma ^*=1/9$ destroys the coherent Kármán vortices and the streamwise vortices over the entire domain displayed. As the twist increases to $\gamma ^*=1/6$ , the upstream vortex coherency is further demolished, but the downstream coherency appears to recover. Further increasing the twist to $\gamma ^*=1/3$ resurges vortex structure coherency for both upstream and downstream. These qualitative observations echo the findings of the non-monotonic effect of twist discussed in §§ 4.1 and 4.2.

Statistical quantification of flow structure coherence level can be made by the POD analysis as introduced in § 3.3. Figure 15( $a$ ) presents the energy fraction of the first 50 POD modes for the ‘whole’ wake region $8D_i \times 6D_i \times L_s$ . Beyond $8D_i$ , the grid resolution of the O-shaped domain is insufficient to support high-fidelity flow simulations. All four cylinders consistently demonstrate energetic classical Kármán vortex shedding in the wake, as evidenced by the high energy fractions of the first two POD modes. The second mode features a quarter wavelength downstream advection of the first mode; see figure 15( $c$ i– $c$ iv), analogous to the wake characteristics behind straight polygonal cylinders (Wang et al. Reference Wang, Gan, Xu and Zhou2020).

Figure 15.

( $a$ ) The POD energy fraction, $\lambda _n / \sum \lambda _n$ , and cumulative energy fraction, $\sum ^n_1 \lambda _n / \sum \lambda _n$ , of the first 50 modes. ( $b$ ) Dependence on $\gamma ^*$ of regional POD cumulative energy fraction. Black squares are for upstream, $0.6D_i\leqslant x\leqslant L_f$ , and red circles for downstream, $L_f\leqslant x \leqslant 8.6D_i$ . ( $c$ i– $c$ iv) The first POD mode of fluctuating transverse velocity ( $v'$ ) in the $x$ – $z$ plane for $\gamma ^* = 0$ , $1/9$ , $1/6$ and $1/3$ , respectively. ( $d$ i) Suboptimal mode 3 of $v'$ for $\gamma ^* = 1/9$ , ( $d$ ii) Suboptimal mode 3 of fluctuating streamwise velocity $u'$ for $\gamma ^* = 1/6$ in the $x$ – $y$ plane at midspan $z=z_0$ , ( $d$ iii) Reconstructed $u^r$ field for $\gamma ^* = 1/6$ using POD mode 3 alone. Dashed lines in ( $c$ ii– $c$ iv) and ( $d$ i) separate RI and RII regimes, cf. figure 9.

The effect of increasing $\gamma ^*$ on the pattern change of the first POD mode reveals that it is not unidirectional. Patterns in the downstream wake region $x\gtrsim 2D_i$ for zero (figure 15 $c$ i) and the strongest twist case (figure 15 $c$ iv) resemble each other, echoing the flow visualisation in figure 14, while the two intermediate twist cases (figure 15 $c$ ii, $c$ iii) display a similar distorted pattern characterised by distinct magnitude distributions. The alternating band-shaped pattern displays a bend towards upstream in central RI, with $\gamma ^*=1/9$ exhibiting a larger magnitude there, whereas $\gamma ^*=1/6$ shows a lower magnitude in central RI, compared with RII, which leads to a broken structure in the near field, $x\lesssim 2D_i$ . The mode magnitude distribution is largely consistent with the velocity spectrum $\varPhi _{vv}$ shown in figure 13( $b$ i– $b$ iv).

As twist is introduced, energy fractions of the first two POD modes appreciably reduce, and energy is distributed towards higher modes. The order of energy fraction of the first two modes does not vary monotonically with $\gamma ^*$ , consistent with the findings discussed above. However, this trend does not align well with that of $\Delta$ OASPL as shown in figure 6( $a$ ). To account for this discrepancy, we now examine the spatial variation of flow coherence by performing POD in upstream and downstream wake regions separately, partitioned at the (spanwise-averaged) vortex formation length $L_f$ , as illustrated in figure 12. Cumulative energy fraction, $\varSigma _1^n \lambda _n$ , for $n=2$ and $5$ , are presented in figure 15( $b$ ) as functions of $\gamma ^*$ .

For the upstream, $x\leqslant L_f$ , figure 15( $b$ ) shows that the coherence level of alternative Kármán vortex shedding ( $n=2$ ) is the highest behind the straight cylinder, as expected, followed by $\gamma ^*=1/3$ . The case associated with the lowest coherence is $\gamma ^*=1/6$ , which plausibly explains its superior noise reduction effect, since the first two modes are responsible for inducing surface pressure fluctuations and radiating acoustic waves (Maryami & Liu Reference Maryami and Liu2024). When more modes are included, the $n=5$ curve indicates that $\gamma ^*=1/9$ and $1/6$ share a similar energy fraction $\approx 40\,\%$ , while the case $\gamma ^*=1/3$ is only marginally higher. On the other hand, in the downstream, $x\geqslant L_f$ , the coherence level for $n=2$ switches between $\gamma ^*=1/9$ and $1/6$ . This lends support to the observation in figure 14 that vortex coherency in the latter tends to recover in the far field with respect to the former. Including more modes to $n=5$ and beyond does not change this trend (see figure 15 $a$ ).

A close inspection of figure 15( $a$ ) reveals that, within the ‘whole’ wake region, modes 3 and 4 of $\gamma ^* = 1/9$ , which appear as a pair, as well as the unpaired mode 3 of $\gamma ^* = 1/6$ , exhibit moderate energy levels, despite substantially lower than the two leading modes. These are suboptimal modes. Mode 3 of $\gamma ^* = 1/9$ is illustrated in figure 15( $d$ i), while mode 4 (not shown) represents a phase-shifted counterpart, consistent with the relationship between mode 2 and mode 1, hence also represents Kármán vortex shedding, but with an appreciably different spatial pattern. This mode exhibits reduced magnitude in RI, in contrast to the distribution seen in the first mode (figure 15 $c$ ii), indicating an RII-dominated shedding mode. Furthermore, spectrum of the leading POD coefficient for mode 3 reveals two peaks of comparable intensity, whereas the spectrum for mode 1 is characterised by a dominant primary peak accompanied by a weaker secondary one (figure not shown). The paired modes 3 and 4 are therefore a manifestation of the two frequencies, $St_1$ and $St_2$ , in RII for $\gamma ^* = 1/9$ , echoing figure 13.

The suboptimal mode 3 of $\gamma ^* = 1/6$ is shown in figure 15( $d$ ii). This distinctive mode exhibits no coherent structure in the central $x$ – $z$ plane, but displays a clear spatial pattern in the $x$ – $y$ plane at $z = z_0$ . This mode appears to represent the time-mean wake, as is more clearly illustrated by the reconstructed streamwise velocity $u^r$ using this mode alone, shown in figure 15( $d$ iii). This type of suboptimal mode – possessing non-trivial energy content yet lacking a paired counterpart – exists only for $\gamma ^* = 1/6$ , which indicates that a larger fraction of the total energy is transferred from the coherent Kármán vortex shedding to this non-shedding mode. The result suggests that the vortex shedding exhibits pronounced intermittency, which is further discussed in the time domain subsequently.

The discussion above indicates the spatial evolution of vortex coherence from a statistical perspective. In the time domain, this originates from temporal variations, i.e. intermittent changes in the vortex shedding pattern from the cylinder surface. This is revealed in figure 16, where time development of $v$ along the span is shown, together with the variation of $C_{\kern-1pt L}$ over the period $280\leqslant t^*\leqslant 430$ . The streamwise sample location is $x=3D_i$ , matching that in figure 13( $b$ i– $b$ iv), where vortex roll-up has completed statistically, as $3D_i\approx L_f$ for $\gamma ^*=1/6$ and $\gt L_f$ for the other cases, see figure 12. Visualisations of 3-D coherent vortex structures at selected time instants are presented in figure 17, where instantaneous flow field is reconstructed using the first five POD modes, according to (3.13), and represented with the $Q$ -criterion isosurface coloured by the normalised spanwise vorticity $\omega _z^{r} D_i/U_{\infty }$ . The superscript $r$ in $\omega _z^{r}$ stands for the POD reconstructed quantity, and $r=5$ . The energy fraction for the sum of the first five modes can be inferred from figure 15( $b$ ).

Figure 16.

Time evolution of ( $a$ i– $d$ i) $C_{\kern-1pt L}$ and ( $a$ ii– $d$ ii) distribution of transverse velocity $v$ along the spanwise line $(3D_i,0,[0,L_s])$ , marked in figure 17(c i), for $\gamma ^* = 0$ , $1/9$ , $1/6$ and $1/3$ , respectively. Dashed lines in ( $b$ ii– $d$ ii) mark the RI/RII partition, cf. figure 9.

Figure 17.

Isosurface of $Q$ -criterion, $Q^r_n \approx 0.09$ , calculated from the velocity field reconstructed using the first five POD modes (for the full field), and coloured by the corresponding reconstructed spanwise vorticity $\omega _z^r D_i/U_{\infty }$ : ( $a$ ) $\gamma ^* = 0$ , ( $b$ i, $b$ ii) $\gamma ^* = 1/9$ , ( $c$ i, $c$ ii) $\gamma ^* = 1/6$ , ( $d$ ) $\gamma ^* = 1/3$ . Black dashed lines in ( $b$ ) and ( $c$ ) mark the RI/RII partition, cf. figure 9, and red dashed curves indicate the vortex core geometry. Similar to figure 14, cylinders in ( $a$ ), ( $c$ ) and ( $d$ ) have been replicated symmetrically at both ends to match the span of the $\gamma ^*=1/9$ case in ( $b$ ). The spanwise probes marked in ( $c$ i) indicate the sample locations for figure 16.

For $\gamma ^* = 0$ , figure 16( $a$ ii) illustrates regular alternative bands, representing a stable and spanwise-uniform vortex shedding pattern. This is supported by the rather constant $C_{\kern-1pt L}$ amplitude over the scrutinised period in figure 16( $a$ i) and the very organised straight vortex tubes shown in figure 17( $a$ ). In contrast, all twisted cylinders demonstrate lower spanwise coherence compared with the straight one. Among the three $\gamma ^*\gt 0$ cases, the strongest twist $\gamma ^*=1/3$ shows a pattern in figure 16( $d$ ii) most similar to the baseline case in figure 16( $a$ ii). The bands remain mostly straight but appear more turbulent, as supported by the increased fluctuation amplitude $C_{\kern-1pt L}'$ shown in figure 16( $d$ i). This is further confirmed in figure 17( $d$ ), where vortex tubes are also straight and organised, yet display a wavy pattern, most pronounced for the near-field vortex tube, reflecting the geometry of the (twisted) cylinder surface. The wavy vortex tube suggests weakened shedding strength, consistent with the observations in Cruikshank, Baba & Lavoie (Reference Cruikshank, Baba and Lavoie2025). The rather organised vortex shedding in these two cases correspond to Pattern I shedding; see figure 5, where a single peak at $St_0$ dominates the far-field noise PSD.

The two intermediate twist cases display appreciably increased vortex shedding intermittency in figure 16( $b$ ii, $c$ ii), also reflected in the behaviour of $C_{\kern-1pt L}$ in figure 16( $b$ i, $c$ i). The case $\gamma ^*=1/9$ is a representative of Pattern II shedding where two dominant peaks are present in the acoustic spectrum (see figure 5), with the lower frequency $St_1$ stemming from regime RI and the higher frequency $St_2$ from RII, as discussed in § 4.2, in particular figure 13. The unmatched shedding frequencies in RI and RII give rise to shedding phase differences – namely, vortex dislocation (Williamson Reference Williamson1992): when the phase mismatch is small, vortex tubes curve upstream, denoted as C-type vortices, in line with the shape of the leading POD mode shown in figure 15( $c$ ii); as the phase mismatch builds up over time, vortex tubes in RI may cross over and connect with the one from the previous cycle in RII, resulting in the less frequent formation of X-type vortices with lower spanwise coherence compared with C-type vortices. These two types of vortex structures are highlighted in figure 17( $b$ i, $b$ ii), and were also observed in previous studies on cylinder and wing wake flows (Darekar & Sherwin Reference Darekar and Sherwin2001; Mittal, Pandi & Hore Reference Mittal, Pandi and Hore2021; Pandi & Mittal Reference Pandi and Mittal2023, Reference Pandi and Mittal2025). Vortex dislocation was previously applied as an effective mechanism for flow and noise control, in the form of distributed forcing (Kim & Choi Reference Kim and Choi2005; Bhattacharya & Gregory Reference Bhattacharya and Gregory2018) and wavy trailing edges (Tombazis & Bearman Reference Tombazis and Bearman1997; Cai, Chng & Tsai Reference Cai, Chng and Tsai2008).

Figure 16( $c$ ii) apparently shows a markedly different pattern, where the RI regime becomes largely dominated by fragmented or broken structures, which are initiated from $t^*\approx 310$ near the midspan of the cylinder and progressively spread towards the two ends over time, with occasional brief recovery of structure coherency, e.g. at $t^*\approx 370$ . At the same time, the amplitude of $C_{\kern-1pt L}$ fluctuation further diminishes from a comparably small value; see figure 16( $c$ i), echoing figure 8( $b$ ). Structure coherency remains discernibly less fragmented in RII in comparison. Noting that the sampling location is at $x\approx L_f$ for this case, these incoherent flow structures detected here are formed upstream and subsequently advected to this point. This is aligned with figure 15( $b$ ) that vortex shedding is of the least coherent Kármán type in the wake behind the cylinder of $\gamma ^*=1/6$ . Figure 17( $c$ i, $c$ ii) presents the vortex structures at two moments – before and after the onset of spanwise coherence loss ( $t^*\approx 310$ ). Vortex structures can also be categorised as C and X types, similar to the $\gamma ^*=1/9$ case, but with a less curved C type appearing occasionally and a more broken X type occurring frequently upstream ( $x\leqslant 3D_i$ ), though subtly reorganised downstream. This intermittency effect will be further discussed in § 4.6.

The cross-correlation of the transverse velocity $v$ along the spanwise direction $(3D_i,0,z)$ is calculated to examine the 3-D coherence in the near wake. The reference point is located at $(3D_i,0,L_s/2)$ . The correlation coefficient between $v(z)$ and $v(z+\Delta z)$ is calculated as

(4.1) \begin{align} R_{v_i v_j}= \frac {\langle v_i(z) \,v_j (z+\Delta z) \rangle }{\sqrt {\langle v^2_i (z)\rangle } \,\sqrt {\langle v^2_j (z+\Delta z)\rangle }}. \end{align}

Figure 18( $a$ ) presents $R_{v_i v_j}$ calculated over a time period corresponding to approximately 30 vortex shedding cycles. For $\gamma ^* = 0$ , $R_{v_i v_j}$ decays slowly with increasing $\Delta z$ , suggesting spanwise-uniform vortex shedding. In contrast, $R_{v_i v_j}$ for the twisted cylinders decay more rapidly, suggesting that twisted configurations reduce the spanwise coherence of vortex shedding, making vortex shedding more 3-D, and thereby contributing to flow and noise control. Among the twisted cases, $\gamma ^* = 1/6$ shows the most significant reduction in spanwise correlation, consistent with its superior noise reduction performance, as shown in figure 6( $a$ ). Additionally, $R_{v_i v_j}$ is recalculated for the time interval before and after $t^*=360$ (see figure 16 $c$ i) to further illustrate its ability to indicate the level of spanwise coherence of the wake structure. The results are shown in figure 18( $b$ ). It is clear that structure fragmentation (cf. $t^*\gt 360$ in figure 16 $c$ ii) significantly reduces the spanwise correlation length, accompanied by the substantially weakened $C_{\kern-1pt L}$ fluctuation (figure 16 $c$ i). As a result, the effective dipole source responsible for the radiated sound is markedly reduced according to Phillips (Reference Phillips1956).

Figure 18.

Spanwise cross-correlation for all the twisted cases in ( $a$ ) and for $\gamma ^* = 1/6$ in ( $b$ ) over two time intervals; cf. figure 16( $c$ i, $c$ ii).

4.4. Low frequency modulation for $\gamma ^* = 1/6$

The optimal twist $\gamma ^* = 1/6$ manifests interesting non-uniform characteristics in both space and time. Spatially, the regional POD analysis shown in figure 15 statistically reveals that the effective noise reduction stems from the low vortex coherency upstream, $x\leqslant L_f$ , echoing figure 18, despite the vortices’ reorganisation downstream. Temporally, figure 16 indicates a strong intermittent vortex shedding behaviour in RI for this case, which is supported quantitatively by Kurtosis (or flatness factor) of $v'$ , i.e. $K(v')=\overline {v'^4}/ (v'_{\textit{rms}} )^4$ , along the sampling line $(3D_i, 0, [0,L_s])$ . Kurtosis is often used to quantify intermittency in a turbulent flow (Davidson Reference Davidson2015). Figure 19( $b$ ) shows that $K(v')$ is distinctively larger for $\gamma ^* = 1/6$ than other twist cases, especially over $0.2\lesssim z/L_s\lesssim 0.8$ , corresponding to RI.

Figure 19.

( $a$ ) The PSD of transverse velocity, $\varPhi _{\mathit{vv}}$ , within the shear layer region at the location $(x_\omega , y_\omega ,0.5L_s)$ , in RI. ( $b$ ) Kurtosis of $v'$ along the line $(3D_i, 0, [0,L_s])$ .

This intermittency can also be detected from the temporal behaviour within the shear layer. Figure 19( $a$ ) presents PSD of $v'$ behind the midspan ( $z/L_s=0.5$ ) within RI. Unlike figure 13( $a$ i– $a$ iv), where the $(x,y)$ coordinates of the sampling location are fixed, here, the probe is placed as per the shedding characteristics of each case. That is, in the streamwise direction, $x_\omega =x_{{sep}}+0.5\overline {L}_{\omega }$ , where $x_{{sep}}$ stands for the $x$ -coordinate of the separation point. This places $x_\omega$ halfway into the time-averaged shear layer which is $\overline {L}_{\omega }$ long, see figure 10. Transversely, it corresponds to the $y$ -coordinate at which the time-averaged maximum $\omega$ magnitude occurs at $x_\omega$ , denoted as $y_\omega$ . Thus, $(x_\omega , y_\omega )$ varies for different cases.

As shown in figure 19( $a$ ), in addition to the K–H instability in the shear layer reflected by the hump within $1\lesssim St\lesssim 4$ for all cases, consistent with figure 13( $a$ i– $a$ iv), the $\gamma ^*=1/6$ case shows another low frequency hump at $St_L$ in RI, which pertains to intermittency, namely modulation of vortex shedding, as shown in figure 21( $c$ ). This low frequency $St_L\approx St_0/6$ (Kiya & Sasaki Reference Kiya and Sasaki1983) has also been observed in the wake flow behind flat plates (Wu et al. Reference Wu, Miau, Hu and Chou2005) and aerofoils (Yang et al. Reference Yang, Pröbsting, Li, Liu and Li2022), which was attributed to low-frequency oscillations of the recirculation bubble.

The configuration of $\gamma ^* = 1/6$ triggers a low-frequency oscillation of the recirculation bubble, which causes modulation in the flow field, leading to broader tones in RI, as shown in figures 13( $a$ iii) and 19( $a$ ), unlike the single tone at $St_1$ in Pattern II ( $\gamma ^* = 1/9$ ). As a result, the far-field noise spectrum exhibits a blunt, broadened peak instead of clearly separated tones (figure 5 $a$ ). In addition, modulation produces a relatively stable region in RI over a specific time period (see figure 21 $c$ ) and a significant disparity in vortex shedding strength between RI and RII, which disrupts spanwise coherence (see figure 18), suppresses the dominant tone, and thus reduces noise emission.

4.5. Acoustic source strengths

The relationship between the characteristics of the near-field wake and far-field noise can be established by examining the acoustic source through the solution of the FW–H equation (3.6). For a rigid stationary surface at low Mach numbers, as discussed in § 3.1, the far-field solution can be simplified as the dipole source term (Curle Reference Curle1955; Glegg & Devenport Reference Glegg and Devenport2017),

(4.2) \begin{align} p'(\boldsymbol{x},t) \approx \frac {x_i}{4\pi |\boldsymbol{x}|^2 c_0} \int _S \left [ \frac {\partial (p_{\textit{ij}} n_{\kern-1pt j})}{\partial \tau }\right ]_{\tau =\tau ^*} \mathrm{d}S, \end{align}

where the far-field observer is located at $\boldsymbol{x}$ , and $\tau ^*=t-|\boldsymbol{x}|/c_0$ denotes the retarded time. Within the compressive stress tensor $p_{\textit{ij}}$ , the contribution of viscous stresses is generally much smaller than that of surface pressure fluctuations and can be neglected in far-field noise prediction (Glegg & Devenport Reference Glegg and Devenport2017). Consequently, the dipole source term in (4.2) is dominated by $p'$ on the cylinder surface, and the resulting far-field sound is approximately proportional to the time derivative, $\mathrm{d}p'/\mathrm{d}t$ . The dipole source strength on the hexagonal cylinder, denoted as $I_p$ , is obtained by first integrating the PSD of the pressure fluctuation derivative $\mathrm{d}p'/\mathrm{d}t$ over the frequency range $0.10 \lesssim St \lesssim 0.21$ , which well covers the dominant tones at $St_0$ , $St_1$ and $St_2$ in figure 5.

Figure 20.

Contours of $I_p$ , normalised by $ U^3_\infty /D_i$ in decibels, over the cylinder surface: ( $a$ ) $\gamma ^* = 0$ , ( $b$ ) $\gamma ^* = 1/9$ , ( $c$ ) $\gamma ^* = 1/6$ , ( $d$ ) $\gamma ^* = 1/3$ . Note the different colourbar ranges. The azimuthal angle $\theta$ was defined in figure 1( $c$ ). Dashed lines in ( $b$ – $d$ ) mark the RI/RII partition, cf. figure 9.

The contours of normalised $I_p$ in figure 20( $a$ ) demonstrate that for the baseline case $\gamma ^* = 0$ , dipole sound sources due to vortex shedding are distributed on the leeward side, with the strongest source strength along the two PSLs $\theta =\pm 60^\circ$ , cf. figure 9( $a$ ), indicating that the vicinity of PSL is the primary noise source (Oguma, Yamagata & Fujisawa Reference Oguma, Yamagata and Fujisawa2013). Applying twist significantly diminishes the $I_p$ level, with its maxima (see the colourbar range) consistent with $\Delta$ OASPL in figure 6( $a$ ). For $\gamma ^*=1/9$ and $1/6$ in figure 20( $b{,}c$ ), high levels of $I_p$ are concentrated around the base area (i.e. $\theta \approx \pm 180^{\circ }$ ) within regime RI, different from the baseline, indicating that shear layer reattachment attenuates the noise source on the twisted cylinder. The $\gamma ^* = 1/6$ case shows the lowest $I_p$ intensity, which is associated with the weakest vortex shedding process, in line with figure 13( $b$ iii). For the strongest twist case $\gamma ^* = 1/3$ , the sound source distribution resembles the baseline case to some extent, with high intensities back to the PSL; see figure 9( $d$ ), suggesting the classical vortex shedding (Pattern I), echoing figure 5( $b$ ).

4.6. Discussion of control mechanisms

The experimental and numerical results shown in § 4.1–§ 4.3 have revealed that applying spanwise twist suppresses far-field noise through modifying the wake flow, though with a nonlinear dependence on the degree of twist, quantified by $\gamma ^*$ . Based on the four simulated cases, the following mechanisms are plausibly valid.

1. (i) A small to moderate twist in the range of $ 1/18 \leqslant \gamma ^* \leqslant 1/7.2$ results in the emergence of dual tones in the noise spectrum with similar intensities substantially lower than that of the straight cylinder, leading to a noise reduction effect. This corresponds to Pattern II of vortex shedding where the mismatch of the two frequencies in RI and RII (figure 13) leads to vortex dislocation, namely the temporally alternating C- and X-type flow structures, as illustrated in figure 21( $b$ ).

2. (ii) Further increasing the twist to $1/5.4\leqslant \gamma ^*\leqslant 1/3$ reduces the number of tones in the noise spectrum back to one (the same for the straight cylinder), corresponding to the classical Kármán vortex shedding, Pattern I. However, the spanwise twist induces wavy vortex tubes, lowering the shedding coherency compared with the straight cylinder, thereby achieving a noise reduction effect that is weaker than Pattern II. This is illustrated in figure 21( $a$ ).

3. (iii) At $\gamma ^* = 1/6$ , or a nearby twist value, the mechanism appears to be a superposition of the two described above, corresponding to a transitional state between Patterns I and II, as illustrated in figure 21( $c$ ). This is supported by the moderate curvature of the C-type structure and the more broken X-type structure, compared with those in Pattern II, see figure 17. Additionally, modulation of vortex shedding in RI further weakens vortex shedding. These result in a blunt peak in the noise spectrum, intermediate between the single tone and dual tones of the two patterns (figure 5), consequently yielding a remarkable and optimal noise reduction effect, especially at high $ \textit{Re}$ (see figure 6).

Figure 21.

Schematic diagram illustrating the flow and noise control mechanisms of the twisted cylinders for ( $a$ ) Pattern I, ( $b$ ) Pattern II and ( $c$ ) transitional pattern ( $\gamma ^* = 1/6$ ).