3.1. Aeroacoustic characteristics

Figure 2 systematically presents the aeroacoustic characteristics of the flow around the circular cylinder under different rotation ratios ( $\alpha = 0, 0.5, 1, 2, 4$ ), revealing the significant influence of forced rotation on both the generation and radiation mechanisms of aerodynamic noise. Figure 2(a) compares the far-field acoustic spectra. The baseline case is dominated by the vortex-shedding tone at $St_0 \approx 0.2$ , together with a higher harmonic near $St_2 \approx 0.6$ . At $\alpha = 0.5$ , the principal spectral component separates into two peaks, $St_{01}\approx 0.18$ and $St_{02}\approx 0.23$ , while the tonal level is substantially reduced. A partial recovery of tonal shedding occurs at $\alpha =1$ , with the dominant frequency shifting to $St\approx 0.24$ . Among the selected conditions, the strongest tonal-noise suppression is obtained at $\alpha =2$ . At $\alpha =4$ , the classical shedding tone is no longer identifiable and is replaced by a low-frequency broadband ‘hump’ centred approximately at $St\approx 0.1$ . These characteristic frequencies are used below to examine the corresponding surface-pressure and wake structures. The one-third-octave-band results in figure 2(b) confirm that the narrowband tonal reduction is accompanied by a broader decrease in the radiated noise, particularly at $ \alpha =2$ . Figure 2(c) further shows that rotation modifies both the magnitude and orientation of the predominantly dipolar radiation pattern. Since the detailed dependence of the far-field level and directivity on the rotation ratio has already been reported by Yang et al. (Reference Yang, Maryami, Liu, Bao, Song and Bi2026), the present study uses these results only as the acoustic reference for the subsequent mechanism analysis.

Figure 2.

The aeroacoustic characteristics of the flow around the circular cylinder under different rotation ratios ( $\alpha = 0, 0.5, 1, 2, 4$ ): ( $ a$ ) acoustic PSD, $\varPhi _{pp}$ , as a function of $St$ at the observation point $(0, 80D, 0)$ ; ( $ b$ ) 1/3-octave-band representation of the acoustic spectra; ( $ c$ ) directivity of the OASPL in the midspan plane ( $z/D = 0$ ). The underlying far-field acoustic signals are from the LES database previously reported by Yang et al. (Reference Yang, Maryami, Liu, Bao, Song and Bi2026).

Figure 3.

Time histories of the lift coefficient $C_l$ , together with the CWT magnitudes of the lift coefficient, $|W_{C_l}|$ , and the corresponding far-field acoustic pressure, $|W_{noise}|$ , for different rotation ratios: (a) baseline, (b) $\alpha =0.5$ , (c) $\alpha =1$ , (d) $\alpha =2$ and (e) $\alpha =4$ . Subpanels (i)–(iii) show $C_l$ , $|W_{C_l}|$ and $|W_{noise}|$ , respectively.

To examine the effects of rotation on the lift and aeroacoustic responses, figure 3 presents the time history of the lift coefficient $C_l$ , together with the CWT magnitudes of the lift coefficient, $|W_{C_l}|$ , and the corresponding far-field acoustic pressure, $|W_{noise}|$ , for the different rotation ratios. Figure 3(a i–e i) show that the lift response evolves from regular periodic oscillations to increasingly irregular fluctuations as the rotation ratio changes. In the baseline case (figure 3 a i), $C_l$ exhibits pronounced periodic variations associated with the classical Kármán vortex street. At $\alpha =0.5$ (figure 3 b i), this periodic behaviour is visibly disturbed. Partial periodicity re-emerges at $\alpha =1$ (figure 3 c i), although the oscillation remains less regular than in the baseline case. At $\alpha =2$ (figure 3 d i), the lift-fluctuation amplitude is substantially reduced. With a further increase to $\alpha =4$ (figure 3 e i), the lift signal becomes irregular and contains pronounced low-frequency variations, indicating departure from the classical periodic shedding regime. The corresponding $|W_{C_l}|$ distributions are shown in figure 3(a ii–e ii). In the baseline case, the CWT magnitude is concentrated near $St\approx 0.2$ , corresponding to the characteristic shedding frequency of the Kármán vortex street. The time–frequency distribution varies non-monotonically with the rotation ratio. At $\alpha =0.5$ and $2$ (figures 3 b ii and 3 d ii, respectively), the CWT magnitude near $St\approx 0.2$ decreases markedly. At $\alpha =1$ (figure 3 c ii), a pronounced frequency component remains near $St\approx 0.24$ , consistent with the partial recovery of periodic shedding. At $\alpha =4$ (figure 3 e ii), the dominant time–frequency content shifts towards a lower-frequency region near $St\approx 0.1$ , indicating the emergence of a rotation-induced low-frequency response. The acoustic-pressure CWT magnitudes, $|W_{noise}|$ , shown in figure 3(a iii–e iii), exhibit trends that broadly correspond to those of $|W_{C_l}|$ . In particular, the suppression, partial recovery and subsequent low-frequency shift of the dominant components occur under the same rotation conditions. This correspondence is consistent with the important role of unsteady lift fluctuations in the predominantly dipolar far-field radiation.

Figure 4.

Real parts of the CWT coefficients of the lift coefficient, $\Re (W_{C_l})$ , and the corresponding far-field acoustic pressure, $\Re (W_{noise})$ , for different rotation ratios: (a) baseline, (b) $\alpha =0.5$ , (c) $\alpha =1$ , (d) $\alpha =2$ and (e) $\alpha =4$ .

Figure 4 further illustrates the time–frequency characteristics of the lift coefficient, $C_l$ , and the far-field acoustic-pressure signal for the different rotation ratios. The real parts of their CWT coefficients are denoted by $\Re (W_{C_l})$ and $\Re (W_{noise})$ , respectively. Two recurring features can be identified in the wavelet fields: quasiperiodic banded patterns and time-localised cone-like patterns. The former appear as approximately horizontal sequences of alternating positive and negative coefficients and remain concentrated around a nearly constant Strouhal number. They are therefore consistent with persistent periodic shedding. In contrast, the cone-like patterns extend over a broader frequency range during limited time intervals, indicating transient broadband spectral content rather than a single persistent frequency component. Their coexistence suggests that the dominant periodic response is accompanied by intermittent broadband fluctuations during the deformation and breakdown of the shear layers (Farge Reference Farge1992). The CWT fields thus distinguish persistent narrowband components from time-localised broadband responses, but do not by themselves establish a physical interscale energy-transfer pathway.

In the baseline case, both $\Re (W_{C_l})$ and $\Re (W_{noise})$ in figure 4(a) exhibit continuous and regularly distributed banded patterns concentrated near $St\approx 0.2$ . This behaviour is consistent with organised large-scale Kármán vortex shedding and the corresponding tonal acoustic response. At $\alpha =0.5$ (figure 4 b), the amplitude of the banded response decreases, while time-localised broadband features become more evident around the principal frequency range. This change is consistent with weakened periodic organisation and an increased contribution from intermittent fluctuations. At $\alpha =1$ (figure 4 c), quasiperiodic banded patterns remain visible, and the dominant frequency shifts slightly upwards to $St\approx 0.24$ , consistent with the partial re-establishment of organised shedding. When the rotation ratio increases to $\alpha =2$ (figure 4 d), the repetitive patterns become sparse and weak, and the time–frequency response is dominated by low-amplitude, time-localised features. This behaviour indicates strong attenuation of the organised large-scale shedding component and a transition towards a weaker and less regularly organised unsteady-loading response. At $\alpha =4$ (figure 4 e), the phase-organised pattern associated with classical vortex shedding is no longer evident. Instead, irregular broadband features appear over a wider range of scales, together with a low-frequency component near $St\approx 0.1$ in both the lift and acoustic signals. These features are consistent with intermittent modulation of the near-wall rotating layer rather than a recovery of the classical Kármán mode. The corresponding spatial structures are examined in the subsequent 3-D analyses.

Beyond the qualitative patterns observed in the time–frequency domain, a quantitative comparison is performed to characterise the non-monotonic dependence of the aeroacoustic response on the rotation ratio $\alpha$ and to examine the corresponding evolution of the time–frequency content. Figure 5(a) compares the root-mean-square (r.m.s.) lift-coefficient fluctuation, $C_{l,{rms}}^{\prime }$ , with the OASPL. The two quantities exhibit closely corresponding non-monotonic trends, consistent with the dominant role of unsteady lift fluctuations in the predominantly dipolar acoustic radiation. Both quantities reach their lowest values among the selected cases at $\alpha =2$ , indicating that this condition provides the strongest suppression of unsteady loading and far-field noise. To quantify the CWT distributions and relate them to the macroscopic fluctuation measures, an integrated wavelet measure, $E$ , is defined from the squared magnitude of the CWT coefficients over the selected time–frequency region. For the lift and far-field acoustic-pressure signals, the corresponding measures are denoted by $E_{C_l}$ and $E_{{noise}}$ , respectively. The measure is defined as

(3.1) \begin{equation} E = \int _{0.05}^{10} \left [ \frac {1}{T} \int _{t_1}^{t_2} \left |W(St,t)\right |^2 \,\mathrm{d}t \right ] \,\mathrm{d}St, \end{equation}

where $W(St,t)$ denotes the CWT coefficient, and $T=t_2-t_1=0.1$ s is the duration of the statistically stationary sampling interval. This quantity is used as a comparative integrated measure of the squared CWT magnitude rather than as a strict physical energy. The variations of $E_{C_l}$ and $E_{{noise}}$ with $\alpha$ are shown in figure 5(b). Their variations are consistent with the macroscopic fluctuation trends and provide a quantitative comparison between the time–frequency responses observed in the CWT fields. At $\alpha =1$ , both measures exhibit a local increase, corresponding to the partial recovery of the organised periodic components shown in figure 4(c), where the CWT magnitude remains concentrated around the characteristic shedding frequency. At $\alpha =2$ , both measures reach their lowest values, consistent with the sparse and weak time–frequency response in figure 4(d) and with the strongest attenuation of unsteady fluctuations among the selected cases. At $\alpha =4$ , the integrated measures increase again owing to the enhanced broadband and low-frequency content. This increase is associated with localised near-wall 3-D unsteadiness rather than with a recovery of the classical spanwise-coherent shedding mode.

Figure 5.

Quantitative comparison of the non-monotonic variation of the unsteady response with rotation ratio $\alpha$ : (a) r.m.s. lift-coefficient fluctuation, $C_{l,{rms}}^{\prime }$ , and OASPL; (b) integrated wavelet measures of the lift coefficient, $E_{C_l}$ , and far-field acoustic pressure, $E_{noise}$ , obtained from the squared CWT magnitude over $0.05\lt St\lt 10$ .

3.2. Near-field pressure analysis

To further reveal the modulation mechanism of forced rotation on the surface pressure distribution, figure 6 systematically presents the contours of the time-averaged pressure coefficient ( $\overline {C}_p$ ) and the r.m.s. of the fluctuating pressure coefficient ( $C_{p,{rms}}^{\prime }$ ) along the cylinder span and circumference for different rotation ratios $\alpha$ . In the baseline case ( $\alpha =0$ ), $\overline {C}_p$ exhibits a typical symmetric distribution. Distinct low-pressure regions form near the separation points of the upper and lower shear layers ( $\theta \in [60^\circ , 80^\circ ]$ and $[280^\circ , 300^\circ ]$ ), while a broad constant-pressure zone in the leeward region ( $[80^\circ , 280^\circ ]$ ) marks the recirculation region characteristic of the baseline separated-flow state. When the rotation ratio increases to $\alpha = 0.5$ , this symmetry is noticeably disrupted. The low-pressure region on the lower side intensifies significantly (reaching a minimum near $\theta \approx 270^\circ$ ), whereas the upper-side suction peak weakens. This asymmetry induces a negative lift, indicating the onset of the `lift crisis’ (Cheng et al. Reference Cheng, Pullin and Samtaney2018). Simultaneously, the recirculation region contracts, marking the initial suppression of regular vortex shedding. As $\alpha$ increases to 1, the upper low-pressure region expands significantly towards the rear, suggesting enhanced boundary layer attachment due to the Magnus effect. Consequently, the lift coefficient transitions from negative to positive, improving aerodynamic performance. At $\alpha =2$ , the asymmetry intensifies: a stable, broad low-pressure region dominates the upper surface, while the lower side transitions to a high-pressure zone. The recirculation region is further compressed and nearly vanishes. Finally, at $\alpha = 4$ , the lower side is completely enveloped by high pressure, while the upper side maintains a broad low-pressure state, indicating that the classical alternating shedding pattern is no longer evident.

Figure 6.

Contours of (i) time-averaged pressure coefficient ( $\overline {C}_p$ ) and (ii) r.m.s. pressure coefficient fluctuations ( $C_{p,{rms}}^{\prime }$ ) at different rotation ratios $\alpha$ : (a) baseline, (b) $\alpha = 0.5$ , (c) $\alpha = 1$ , (d) $\alpha = 2$ , (e) $\alpha = 4$ .

The distribution of $C_{p,{rms}}^{\prime }$ further elucidates the evolution of unsteady wall pressure fluctuations. For the baseline, peak fluctuation intensities occur near the separation points ( $\theta \approx 70^\circ$ and $290^\circ$ ), driven by the oscillating shear layers. A striking phenomenon emerges at $\alpha = 0.5$ . While the fluctuation intensity on the upper side diminishes due to stabilisation, the lower side ( $\theta \approx 270^\circ$ ) exhibits a distinct high-intensity band of fluctuations, exceeding even the baseline levels. Although this suggests strong local unsteadiness, it contrasts with the global noise reduction observed in far-field spectra. This implies that while the lower shear layer sheds vigorously near the wall, the resulting structures are highly localised and lack the spanwise coherence necessary for efficient noise radiation. As $\alpha$ increases to 2, the upper separation point shifts downstream with low fluctuation levels, and the lower-side fluctuations are effectively suppressed, indicating strong attenuation of the dominant shedding-related wake fluctuations. Finally, at $\alpha =4$ , high-intensity pressure fluctuations re-emerge and extend over a broad region of the cylinder surface. These irregular patterns are consistent with the localised near-wall 3-D unsteadiness identified later from the instantaneous vortex and DMD fields. The relatively uniform spanwise distributions of $\overline {C}_p$ and $C_{p,{rms}}^{\prime }$ indicate approximate spanwise homogeneity of the first- and second-order pressure statistics. This statistical uniformity does not, however, imply spanwise coherence of the instantaneous or frequency-resolved flow structures.

Figure 7.

Time–space distribution of the surface pressure coefficient ( $C_p$ ) on the cylinder under different rotation ratios $\alpha$ : (a) baseline, (b) $\alpha = 0.5$ , (c) $\alpha = 1$ , (d) $\alpha = 2$ , (e) $\alpha = 4$ .

To further explore the impact of forced rotation on the time–space distribution characteristics of the cylinder surface pressure, figure 7 presents the time–space evolution of the surface pressure coefficient $C_p$ under different rotation ratios. In the baseline case, the pressure time–space map exhibits a regular periodic stripe structure, corresponding to the vortex shedding frequency of approximately 210 Hz, which is consistent with the frequency spectrum analysis results shown in figure 2(a). It is evident that the pressure fluctuations on the upper and lower surfaces of the cylinder are out of phase, reflecting the alternating shedding of vortices on the top and bottom sides of the cylinder, eventually evolving into the typical Kármán vortex street in the wake. Clear low-pressure fluctuation zones are observed at $\theta \approx 70^\circ$ and $290^\circ$ , indicating the development of shear layer instability. When the rotation ratio increases to $\alpha = 0.5$ , the flow characteristics on the cylinder surface change significantly: the upper-side low-pressure fluctuation zone shifts towards $\theta \approx 75^\circ$ , and the vortex shedding process exhibits dual-frequency components, which closely align with the ‘dual-peak’ phenomenon revealed in figure 2(a). Meanwhile, the lower-side low-pressure fluctuation zone around $\theta \approx 285^\circ$ shows intense non-stationarity, characterised by the presence of multiple sharp peaks in the pressure signal, indicating enhanced wall pressure fluctuations.

As the rotation ratio further increases to $\alpha = 1$ , the upper-side low-pressure zone ( $\theta \approx 80^\circ$ ) regains a stable and clear periodic feature, with a significant expansion of the fluctuation range, indicating the recovery of vortex shedding behaviour. Although the pressure fluctuation near $\theta \approx 280^\circ$ on the lower side increases slightly, the sharp peak-like irregular disturbances are significantly reduced, suggesting that the local pressure response becomes more regular. Under the condition of $\alpha = 2$ , the upper-side low-pressure zone ( $\theta \approx 90^\circ$ ) tends towards a steady distribution, and the pressure fluctuations further weaken. On the lower side ( $\theta \approx 270^\circ$ ), $C_p$ shifts towards higher values, while the periodic pressure pattern progressively weakens, consistent with the suppression of coherent vortex shedding. Finally, under the condition of $\alpha = 4$ , the pressure fluctuation regions at $\theta \approx 90^\circ$ exhibit partial periodicity, corresponding to flow structures at approximately 125 Hz ( $ St\approx 0.1$ ), which is consistent with the conclusions drawn from figures 2 to 4.

Figure 8.

Spanwise distributions of the pressure power spectral density ( $\varPhi _{pp,s}$ ) as a function of $St$ at different rotation ratios for three typical circumferential angles, $\theta =90^\circ$ , $180^\circ$ , and $270^\circ$ . Panels (a)–(e) correspond to the baseline case and rotation ratios $\alpha =0.5$ , $1$ , $2$ and $4$ , respectively. Subpanels (i) to (iii) correspond to (i) $\theta =90^\circ$ (upper side), (ii) $\theta =180^\circ$ (leeward side) and (iii) $\theta =270^\circ$ (lower side), respectively.

Figure 8 shows the spanwise pressure PSD ( $\varPhi _{pp,s}$ ) as a function of $St$ at three circumferential positions on the cylinder surface ( $\theta = 90^\circ$ , $180^\circ$ and $270^\circ$ ) for different rotation ratios. In the baseline case (figure 8 a), the spectral structure clearly exhibits the typical periodic vortex shedding characteristics. At both $\theta = 90^\circ$ and $\theta = 270^\circ$ , strong and spanwise-consistent spectral peaks appear near the dominant frequency $St_0 \approx 0.2$ , indicating that large-scale coherent vortices alternate shedding symmetrically from the upper and lower shear layers. In the centre of the leeward side at $\theta = 180^\circ$ , the spectrum is dominated by the first harmonic component at $St_1 \approx 0.4$ , reflecting the characteristic disturbance frequency of the recirculating vortex. As the rotation ratio increases to $\alpha = 0.5$ (figure 8 b), the spectral features undergo significant changes: the original main frequency $St_0$ splits into two new characteristic frequencies, $St_{01}$ and $St_{02}$ , which exhibit discontinuous distributions in the spanwise direction, disrupting the original spanwise coherence. This phenomenon indicates that the dominant pressure response separates into two frequency components, accompanied by reduced spanwise organisation, effectively reducing the coherence and intensity of the far-field sound source, thereby weakening the noise radiation. This dual-frequency coexistence phenomenon is in good agreement with the dual-peak noise spectrum observed in figure 2. Notably, at $\theta = 270^\circ$ , the spectral peak intensity is significantly higher than at the other observation points, further confirming the trend of increased lower surface pressure fluctuations and intensified non-stationarity shown in figure 6.

When the rotation ratio increases to $\alpha = 1$ (figure 8 c), the spectral characteristics further evolve: at $\theta = 90^\circ$ , the dominant frequency peak $St_0$ remains and its energy intensity significantly increases compared with $\alpha = 0.5$ ; however, the single-tone spectral peaks at $\theta = 180^\circ$ and $\theta = 270^\circ$ almost disappear, replaced by low-energy, widely distributed broadband structures. This phenomenon explains the higher overall noise level observed at $\alpha = 1$ in figure 2, where the pressure fluctuation enhancement in the upper separation zone and the recovery of vortex shedding contribute to the dominant far-field sound radiation. At $\alpha = 2$ (figure 8 d), the spectral energy at all three observation points further decreases. Only a faint spectral peak remains at $\theta = 90^\circ$ , with the dominant frequency almost completely erased, indicating the significant suppression of periodic vortex shedding mechanisms. Finally, when the rotation ratio increases to $\alpha = 4$ (figure 8 e), the narrowband signature of the classical Kármán mode is no longer evident. The spectra become predominantly broadband, with a low-frequency elevation near $St\approx 0.1$ . This response indicates that the classical alternating shedding process has been replaced by localised, non-periodic 3-D disturbances in the near-wall rotating layer. This broadband response characteristic is consistent with the high-amplitude pressure fluctuation phenomenon observed in figure 6 and the broadband noise characteristics in figure 2.

To distinguish convective wall-pressure fluctuations from low-wavenumber components associated with spatially synchronous surface loading, a wavenumber–frequency spectrum (WFS) analysis is performed (Corcos Reference Corcos1964; Choi & Moin Reference Choi and Moin1990; Abraham & Keith Reference Abraham and Keith1998; Zhao et al. Reference Zhao, Shi, Zhang, Chen and Zhang2026). For each half of the cylinder, the fluctuating surface pressure is sampled at $N_s=37$ equally spaced circumferential locations, with a spatial interval of $\Delta s=\pi D/72$ . On both halves, the positive surface coordinate is defined from the upstream stagnation point towards the rear stagnation region. The temporal cross-PSD between the surface-pressure signals at two locations, $s_i$ and $s_{\!j}$ , is first estimated using Welch’s method and is denoted by $G_{ij}(\omega _f)$ , where $\omega _f=2\pi f$ is the angular frequency. For a given spatial separation $m\Delta s$ , the cross-spectral density is averaged over all available sampling-point pairs having the same separation:

(3.2) \begin{equation} \overline {G}_{pp}(m\Delta s,\omega _f) = \frac {1}{N_s-m} \sum _{j=1}^{N_s-m} G_{\!j,j+m}(\omega _f), \qquad m=0,1,\ldots ,N_s-1. \end{equation}

The WFS is then obtained by applying a spatial Fourier transform to the separation-dependent cross-spectral density,

(3.3) \begin{equation} \varPhi _{pp}(k_s,\omega _f) = \frac {1}{2\pi } \int _{-\infty }^{+\infty } \overline {G}_{pp}(\xi ,\omega _f) \exp (-\mathrm{i}k_s\xi ) \,\mathrm{d}\xi , \end{equation}

where $\xi$ denotes the circumferential separation between two sampling locations.

In the discrete implementation, this expression is evaluated as

(3.4) \begin{equation} \begin{aligned} \varPhi _{pp}(k_s,\omega _f) \approx \frac {\Delta s}{2\pi } \Bigg [ &\overline {G}_{pp}(0,\omega _f) \\ &+ \sum _{m=1}^{N_s-1} \Big ( \overline {G}_{pp}(m\Delta s,\omega _f) \mathrm{e}^{-\mathrm{i}k_s m\Delta s} + \overline {G}_{pp}(-m\Delta s,\omega _f) \mathrm{e}^{\mathrm{i}k_s m\Delta s} \Big ) \Bigg ]. \end{aligned} \end{equation}

Here, the positive- and negative-separation cross-spectra correspond to the two possible orderings of each sampling-point pair. For real-valued pressure signals, they satisfy the Hermitian relation $\overline {G}_{pp}(-m\Delta s,\omega _f)=\overline {G}_{pp}^{\,*}(m\Delta s,\omega _f)$ . The cross-spectral densities are calculated using a Hanning window with $2^{14}$ samples and $50\,\%$ overlap. The resulting spectrum describes the distribution of the surface-pressure fluctuations over angular frequency $\omega _f$ and circumferential wavenumber $k_s$ .

The WFS characterises the distribution of wall-pressure fluctuation energy over temporal and spatial scales. A sloping spectral ridge represents pressure disturbances convecting along the cylinder surface, and its slope provides an estimate of the surface convection velocity $ U_c \approx {\omega _f}/{k_s}$ . In contrast, spectral components concentrated near $k_s=0$ correspond to pressure fluctuations that vary slowly along the surface and are therefore spatially synchronous over an extended circumferential region (Farabee & Casarella Reference Farabee and Casarella1991; Kang & Lee Reference Kang and Lee2025). Because the present LES is incompressible, the WFS is not used to identify a propagating acoustic dispersion branch. Instead, it is employed to distinguish convective hydrodynamic pressure fluctuations from low-wavenumber components associated with coherent surface loading.

Figure 9 presents the WFS of the surface-pressure fluctuations on the upper-half ( $0^\circ$ – $180^\circ$ , figure 9 i) and the lower-half ( $360^\circ$ – $180^\circ$ , figure 9 ii) of the cylinder. In the baseline case (figure 9 a), both surfaces exhibit distinct high-energy convective ridges, whose slopes correspond approximately to $U_c\approx 0.6U_\infty$ . These ridges are consistent with organised pressure disturbances convecting within the separated shear layers. A pronounced low-wavenumber component is also observed near $St\approx 0.2$ , indicating spatially coherent surface loading associated with the natural vortex-shedding mode. At $\alpha =0.5$ and $1$ (figures 9 b and 9 c ,respectively), the upper–lower symmetry is substantially modified. The convective ridge on the upper surface weakens, whereas the lower surface retains appreciable spectral energy. This behaviour is consistent with the elevated $C_{p,{rms}}^{\prime }$ on the lower side shown in figure 6(b,c), indicating that the lower shear layer remains dynamically active. At $\alpha =0.5$ , the low-wavenumber energy is also distributed between the two characteristic frequency components, consistent with the frequency splitting and loss of spanwise organisation identified from the surface-pressure and wake spectra. At $\alpha =1$ , part of the organised convective and low-wavenumber response reappears, in agreement with the partial recovery of tonal shedding. At $\alpha =2$ (figure 9 d), both the convective ridges and the low-wavenumber pressure components are substantially attenuated on the upper and lower surfaces. This result indicates a simultaneous weakening of convective shear-layer disturbances and spatially synchronous surface loading. Together with the reduced pressure r.m.s., wake-velocity fluctuations and spanwise coherence, the WFS provides further evidence that the strong noise reduction at this condition results from both lower loading amplitude and diminished spatial organisation. Finally, at $\alpha =4$ (figure 9 e), the WFS becomes predominantly broadband and the classical shedding-related ridge is no longer evident. A low-wavenumber component remains near $St\approx 0.1$ , indicating spatially extended low-frequency surface loading. The negative- $k_s$ content on the lower surface suggests oppositely directed phase propagation or locally reversed surface-convective motion within the near-wall rotating layer. These spectral features are consistent with the localised streamwise and helical structures identified in the subsequent 3-D analyses, whereas the downstream wake remains comparatively weak.

Figure 9.

Wavenumber–frequency spectra of the surface-pressure fluctuations, $\varPhi _{pp}(k_s,\omega _f)$ . The colour scale represents the spectral level, $10\log _{10}[\varPhi _{pp}(k_s,\omega _f)/p_{{ref}}^2]$ . Subpanels (i) corresponds to the upper-half of the cylinder ( $0^\circ$ – $180^\circ$ ), and (ii) corresponds to the lower-half ( $360^\circ$ – $180^\circ$ ). On both halves, the positive surface coordinate $s$ is defined from the upstream stagnation point towards the rear stagnation region. The dashed lines indicate the dispersion relations associated with the estimated convection velocity $U_c$ and the free stream velocity $U_\infty$ . Panels (a)–(e) correspond to the baseline case and rotation ratios $\alpha =0.5$ , $1$ , $2$ and $4$ , respectively. The horizontal axis denotes the dimensionless surface wavenumber $k_sD$ , and the vertical axis denotes the Strouhal number $St$ .

3.3. Wake velocity analysis

To characterise the 3-D organisation of the wake fluctuations, figure 10 presents the r.m.s. streamwise and transverse velocity fluctuations in the central $x$ – $y$ plane and in the streamwise–spanwise $x$ – $z$ plane at $y=0$ . The corresponding $x$ – $y$ -plane distributions and their variation over a wider range of rotation ratios were reported previously in Yang et al. (Reference Yang, Maryami, Liu, Bao, Song and Bi2026). They are retained here as a reference for the newly presented $x$ – $z$ -plane distributions, which are used to examine the spanwise organisation of the wake. In the baseline case, the near wake exhibits pronounced velocity fluctuations, with $v'_{{rms}}$ dominating over $u'_{{rms}}$ in the region $x/D\lt 3$ , consistent with periodic vortex shedding. The $x$ – $z$ -plane distributions show nearly uniform high-intensity bands along the span, indicating a strongly organised wake. At $\alpha =0.5$ , the high-intensity regions contract and shift downstream, while their spanwise uniformity is weakened. A partial recovery of the transverse fluctuations occurs at $\alpha =1$ , consistent with the re-emergence of tonal shedding. At $\alpha =2$ , both the in-plane fluctuation levels and the spanwise wake signatures are substantially reduced. In particular, the disappearance of the spanwise-uniform high-intensity bands indicates that the suppression of the wake fluctuations is accompanied by a loss of large-scale spanwise organisation. At $\alpha =4$ , enhanced fluctuations reappear locally near the cylinder, whereas the downstream wake remains weaker than in the baseline case. The localised high-rotation response differs from the spanwise-organised vortex-shedding pattern of the stationary cylinder and is examined further through the frequency-resolved and 3-D analyses below.

Figure 10.

Contours of the r.m.s. streamwise ( $u'_{{rms}}/U_{\infty }$ ) and transverse ( $v'_{{rms}}/U_{\infty }$ ) velocity fluctuations in the central $x$ – $y$ plane (i and ii) and the streamwise–spanwise $x$ – $z$ plane at $y=0$ (iii and iv). The rotation ratios are (a) baseline, (b) $\alpha =0.5$ , (c) $\alpha =1$ , (d) $\alpha =2$ and (e) $\alpha =4$ . The $x$ – $y$ -plane distributions were previously reported by Yang et al. (Reference Yang, Maryami, Liu, Bao, Song and Bi2026); the $x$ – $z$ -plane distributions are presented here to examine the spanwise organisation of the wake fluctuations.

Figure 11 shows the time-averaged spanwise-vorticity distribution of the flow around the cylinder under different $\alpha$ . The spanwise vorticity is defined as $\overline {\omega }_z = \partial \overline {v}/\partial x - \partial \overline {u}/\partial y$ , and is presented in its non-dimensional form $\overline {\omega }_z D / U\infty$ to reveal the overall evolution characteristics of the shear layer and wake structure. In the baseline case (figure 11 a), the mean upper and lower shear layers exhibit an approximately symmetric distribution and extend into the near wake. This mean organisation is consistent with the alternating shedding identified from the instantaneous and spectral results. As the rotation ratio increases to $\alpha = 0.5$ (figure 11 b), the symmetry of the flow field is significantly disrupted. The upper shear layer separation point shifts downstream, and a stretching and stabilisation trend is observed, with the shedding position delayed to $x/D \approx 2$ . This indicates that rotation weakens the interaction between the upper flow and the wall. In contrast, the lower shear layer becomes unstable and sheds earlier, showing stronger non-stationarity.

As the rotation ratio increases further to $\alpha = 1$ and $\alpha = 2$ (figures 11 c and 11 d, respectively), the separation point of the upper shear layer continues to move downstream, with the length of the shear layer shortening, and a clear clockwise deflection appears. Meanwhile, the strength of the lower shear layer decreases significantly, and the shedding process is effectively suppressed, indicating that the rotation’s control over the vortex shedding mechanism is becoming stronger. When the rotation ratio reaches $\alpha = 4$ (figure 11 e), strong mean vorticity is concentrated within a near-wall rotating layer, while discrete alternating vortex cores are no longer evident in the downstream wake. This indicates suppression of the classical Kármán shedding mode and a comparatively weak downstream wake. The flow is nevertheless not completely steady, because localised 3-D disturbances persist near the cylinder and are associated with the low-frequency broadband response observed in figure 2.

Figure 11.

Time-averaged spanwise-vorticity contours under different rotation ratios, non-dimensionalised by $\overline {\omega }_z D / U\infty$ : ( $ a$ ) baseline, ( $ b$ ) $\alpha = 0.5$ , (c) $\alpha = 1$ , ( $ d$ ) $\alpha = 2$ , ( $ e$ ) $\alpha = 4$ .

Figure 12.

Comparison of velocity PSD distributions along the spanwise direction at the streamwise location $x/D = 1$ under different rotation ratios: ( $ a$ ) baseline, ( $ b$ ) $\alpha = 0.5$ , ( $ c$ ) $\alpha = 1$ , ( $ d$ ) $\alpha = 2$ , ( $ e$ ) $\alpha = 4$ . Panels (a) to (e) correspond to one rotation ratio condition: (i) shows the PSD of the streamwise velocity $u$ ( $\varPhi _{uu}$ ), and (ii) shows the PSD of the transverse velocity $v$ ( $\varPhi _{vv}$ ).

To further investigate the control mechanism of forced rotation on the wake turbulence structure, figure 12 shows the PSD distributions of the streamwise velocity component ( $u$ ), $\varPhi _{uu}$ , and the transverse velocity component ( $v$ ), $\varPhi _{vv}$ , measured at $x/D = 1$ and $y = 0$ for different rotation ratios. In the baseline case, the frequency spectrum of $\varPhi _{vv}$ is significantly higher than that of $\varPhi _{uu}$ , and exhibits a distinct peak at $St_0 \approx 0.2$ , corresponding to the typical vortex shedding frequency. This shows that the baseline wake dynamics are dominated by transverse velocity fluctuations associated with the natural shedding mode. At $\alpha = 0.5$ , the spectrum of $\varPhi _{vv}$ develops a bimodal structure with two dominant frequencies, $St_{01} \approx 0.18$ and $St_{02} \approx 0.23$ , consistent with the patterns in figures 2 and 8. This indicates the coexistence of two unsteady shedding modes. As the rotation ratio increases to $\alpha = 1$ , the main spectral peak of $\varPhi _{vv}$ persists with slightly higher intensity than at $\alpha = 0.5$ , suggesting partial recovery of unsteady vortex shedding and a corresponding reduction in noise suppression, consistent with figure 2. At $\alpha = 2$ , the spectral intensity of both $\varPhi _{uu}$ and $\varPhi _{vv}$ drops sharply, with the energy near $St_0$ nearly disappearing. This indicates effective suppression of periodic shedding and unsteady wake disturbances, leading to a less energetic and less organised wake. At $\alpha = 4$ , $\varPhi _{vv}$ shows localised enhancement at low frequencies ( $St \lt 0.1$ ), concentrated in $z/D = -2 \sim -0.5$ and $z/D = 0.5 \sim 2$ . This suggests that, although large-scale periodic structures are suppressed, strong low-frequency disturbances remain. These correspond to the low-frequency hump observed in figure 2, indicating the emergence of localised low-frequency disturbance components that may affect wake stability and acoustic radiation.

To further quantify the effect of rotation on the spanwise organisation of the wake fluctuations, the frequency-dependent spanwise coherence length, $\varLambda _{vv}(St)$ , is evaluated using the transverse velocity component $v$ . The radiation efficiency of a spanwise-distributed loading source is strongly influenced by its spanwise coherence. Therefore, a reduction in the coherence length is generally associated with less efficient far-field acoustic radiation (Amiet Reference Amiet1975; Kato et al. Reference Kato, Iida, Takano, Fujita and Ikegawa1993; Seo & Moon Reference Seo and Moon2005). Two transverse-velocity signals are considered at the same streamwise and transverse location, $(x/D,y/D)=(1,0)$ , but separated by a spanwise distance $\Delta z$ . The signals are denoted by $v_0(t)=v(x,y,z_0,t)$ and $v_{\Delta }(t)=v(x,y,z_0+\Delta z,t)$ . Their magnitude-squared coherence is defined as

(3.5) \begin{equation} C_{vv}(St,\Delta z) = \frac { \left | \varPhi _{v_0v_{\Delta }}(St,\Delta z) \right |^2 }{ \varPhi _{v_0v_0}(St) \varPhi _{v_{\Delta }v_{\Delta }}(St) }, \end{equation}

where $\varPhi _{v_0v_{\Delta }}$ is the cross-PSD between the two velocity signals, while $\varPhi _{v_0v_0}$ and $\varPhi _{v_{\Delta }v_{\Delta }}$ are their respective auto-PSD.

The frequency-dependent spanwise coherence length is then defined as

(3.6) \begin{equation} \varLambda _{vv}(St) = \int _{0}^{L_z/2} C_{vv}(St,\Delta z) \,\mathrm{d}(\Delta z). \end{equation}

The upper integration limit $L_z/2$ corresponds to the maximum independent spanwise separation in the periodic computational domain. The velocity-based coherence length is used here as a hydrodynamic indicator of the spanwise organisation of the wake rather than as a direct measure of acoustic-source coherence.

Figure 13 shows that, in the baseline case, the spanwise coherence length exhibits a pronounced peak of $\varLambda _{vv}/D\approx 1.8$ near the dominant vortex-shedding frequency ( $St\approx 0.2$ ). This feature is consistent with the presence of highly organised, large-scale wake structures with strong spanwise correlation, which are associated with the pronounced narrowband acoustic response. With the application of rotation, this characteristic length scale decreases markedly. At $\alpha =0.5$ , although the near-field analysis (figure 6) indicates intensified pressure fluctuations on the lower surface, figure 13 shows that $\varLambda _{vv}$ decreases substantially relative to the baseline. Since the radiation efficiency of a spanwise-distributed loading source is influenced by its spanwise correlation length (Amiet Reference Amiet1975), the combined evidence indicates that the loss of spanwise organisation reduces the efficiency with which distributed loading fluctuations combine in the far field. This provides a physical explanation for the reduced tonal radiation despite the locally enhanced pressure fluctuations. For the strongest noise-reduction case, $\alpha =2$ , the coherence length reaches its minimum among the investigated conditions over the principal shedding-frequency range. The reduction in fluctuation amplitude is therefore accompanied by a marked loss of large-scale spanwise organisation. This behaviour is consistent with the breakup of the organised shedding-related structures and the resulting decrease in far-field acoustic radiation efficiency.

To visually elucidate the impact of forced rotation on the multiscale turbulent topological structures of the cylinder wake, the $\varOmega$ vortex identification method proposed by Liu et al. (Reference Liu, Wang, Yang and Duan2016) is employed to extract 3-D coherent structures. In contrast to traditional second-generation vortex identification methods (e.g. $Q$ , $\lambda _2$ and $\Delta$ criteria), the primary advantage of the $\varOmega$ method lies in its ability to not only reduce the sensitivity to threshold selection but also effectively identify vortex structures of varying intensities, thereby preventing strong shear from masking weak vortex signals. The $\varOmega$ criterion is defined based on the symmetric part (strain rate tensor $\boldsymbol{A}$ ) and the antisymmetric part (vorticity tensor $\boldsymbol{B}$ ) of the velocity gradient tensor, calculated as

(3.7) \begin{equation} \varOmega = \frac {\|\boldsymbol{B}\|_F^2}{\|\boldsymbol{A}\|_F^2 + \|\boldsymbol{B}\|_F^2 + \varepsilon } \end{equation}

where $\|\boldsymbol{\cdot }\|_F$ denotes the Frobenius norm, and $\varepsilon$ is a small parameter introduced to prevent division by zero. The value of $\varOmega$ is normalised to the range $[0, 1]$ . Following the recommendation of Liu et al. (Reference Liu, Wang, Yang and Duan2016), an isosurface of $\varOmega =0.52$ is selected in this study to visualise the vortex structures, which are coloured by the dimensionless streamwise vorticity $\omega _x D/U_{\infty }$ to indicate the sense of rotation. Figure 14 illustrates the evolution of vortex structures from the upper and lower perspectives of the cylinder under different rotation ratios. For the baseline case, the wake exhibits typical characteristics of a Kármán vortex street, featuring staggered large-scale spanwise rollers interconnected by streamwise ribs. With the rotation imposed, the flow topology undergoes drastic changes. As the rotation ratio increases, the shedding of large-scale spanwise vortices is significantly suppressed; the wake structure gradually fragments and becomes dominated by asymmetric small-scale turbulence. Notably, at the high rotation ratio ( $\alpha =4$ ), as shown in figure 14(e), the classical alternating vortex-shedding pattern is no longer evident. Instead, Taylor–Görtler-like helical structures develop within the near-wall rotating layer, consistent with the observations of Aljure et al. (Reference Aljure, Rodríguez, Lehmkuhl, Pérez-Segarra and Oliva2015). Their spatial scale and organisation suggest that they are associated with, and may provide a hydrodynamic origin for, the low-frequency elevation near $St\approx 0.1$ in the pressure and acoustic spectra.

Figure 13.

Frequency-dependent spanwise coherence length, $\varLambda _{vv}/D$ , based on the magnitude-squared coherence of the transverse velocity fluctuations at $x/D=1$ and $y/D=0$ for the baseline case and rotation ratios $\alpha =0.5$ , $1$ , $2$ and $4$ .

Figure 14.

Isosurfaces of instantaneous 3-D vortex structures identified by the $\varOmega$ criterion ( $\varOmega =0.52$ ) for varying rotation ratios: ( $ a$ ) baseline, ( $ b$ ) $\alpha =0.5$ , ( $ c$ ) $\alpha =1$ , ( $ d$ ) $\alpha =2$ and ( $ e$ ) $\alpha =4$ . Subpanels (i) and (ii) display the upper side and lower side views, respectively. The structures are coloured by the dimensionless streamwise vorticity $\omega _x D/U_{\infty }$ (red for positive, blue for negative). Note the emergence of Taylor-Görtler-like helical vortices wrapping around the cylinder surface at $\alpha =4$ .

Figure 15.

Spatiotemporal evolution of the backward-time finite-time Lyapunov exponent (FTLE) fields, $\sigma (\boldsymbol{x},t)$ , evaluated in the midspan $x$ – $y$ plane. Panels (a)–(e) correspond to rotation ratios $\alpha =0$ , $0.5$ , $1$ , $2$ and $4$ , respectively. Subpanels (i) to (iv) represent consecutive time instants $t_0$ , $t_0+\Delta t$ , $t_0+2\Delta t$ and $t_0+3\Delta t$ . The magnitude of the backward integration interval is $|T|=6.5D/U_\infty$ . High-value ridges indicate attracting material lines and the principal in-plane transport pathways associated with the separated shear layers. The white dashed circles track the formation and roll-up of large-scale spiral lobes.

Figure 15. Long description

Panel A: A series of heat maps showing the evolution of FTLE fields at different time instants. The x-axis is labeled x/D and the y-axis is labeled y/D. The color scale on the right indicates FTLE values in units of 10³ s^-1. Subpanels (i) to (iv) represent consecutive time instants t0, t1, t2, and t3. High-value ridges indicate attracting material lines and principal in-plane transport pathways. White dashed circles track the formation and roll-up of large-scale spiral lobes. Panel B: Similar heat maps for a different rotation ratio. Panel C: Heat maps for another rotation ratio. Panel D: Heat maps for yet another rotation ratio. Panel E: Heat maps for the final rotation ratio.

While the preceding Eulerian analysis of the velocity and vorticity fields provides insight into the flow dynamics, it does not directly describe the unsteady material-transport pathways within the wake. Lagrangian coherent structure (LCS) analysis is therefore employed to characterise the finite-time organisation of fluid–particle trajectories. Unlike conventional Eulerian measures, the LCS framework is objective under time-dependent rigid-body transformations and can be used to identify material structures that organise fluid transport (Haller Reference Haller2005; Peacock & Haller Reference Peacock and Haller2013). In the present study, LCSs are extracted from the FTLE, which quantifies the finite-time exponential separation of initially neighbouring fluid–particle trajectories (Haller Reference Haller2002; Shadden et al. Reference Shadden, Lekien and Marsden2005; Duan & Wang Reference Duan and Wang2024). The flow map $\phi _{t_0}^{t_0+T}:\boldsymbol{x}(t_0)\mapsto \boldsymbol{x}(t_0+T)$ maps a particle from its initial position at $t_0$ to its position at $t_0+T$ . From the deformation gradient of the flow map, $\boldsymbol{\nabla }\phi _{t_0}^{t_0+T}$ , the right Cauchy–Green deformation tensor is constructed as

(3.8) \begin{equation} \boldsymbol{C}(\boldsymbol{x},t_0,T) = \left [ \boldsymbol{\nabla }\phi _{t_0}^{t_0+T}(\boldsymbol{x}) \right ]^{\top } \boldsymbol{\nabla }\phi _{t_0}^{t_0+T}(\boldsymbol{x}). \end{equation}

This tensor characterises the finite-time stretching of material elements. The FTLE field is then obtained from the largest eigenvalue of $\boldsymbol{C}$ :

(3.9) \begin{equation} \sigma (\boldsymbol{x},t_0,T) = \frac {1}{|T|} \ln \sqrt {\lambda _{\textit{max}}\!\left (\boldsymbol{C}\right )}. \end{equation}

Because the present analysis focuses on wake morphology and entrainment pathways, backward-time integration ( $T\lt 0$ ) is employed. Particle trajectories are integrated using a fourth-order Runge–Kutta scheme, with an integration duration of $|T|=6.5D/U_\infty$ . The spatial gradient of the flow map is evaluated on a high-resolution Cartesian grid using central differences, while cubic-spline interpolation is used to reconstruct the velocity field along the particle trajectories. The FTLE field is evaluated in the midspan $x$ – $y$ plane and therefore characterises in-plane material transport. Out-of-plane transport is not represented by the present planar LCS analysis. Under this configuration, high-value ridges of the backward-time FTLE field identify attracting material lines (Shadden et al. Reference Shadden, Lekien and Marsden2005; Green et al. Reference Green, Rowley and Haller2007). These ridges delineate the principal material-transport structures associated with the separated shear layers and indicate the pathways through which free stream fluid is entrained into the wake (Cui & Feng Reference Cui and Feng2025).

Figure 15 shows the temporal evolution of the backward-time FTLE fields for the different rotation ratios ( $\alpha =0$ , $0.5$ , $1$ , $2$ and $4$ ). In the baseline case ( $\alpha =0$ , figure 15 a), strong attracting ridges originating from the upper and lower separation regions roll up alternately. As indicated by the white dashed circles, these ridges form large-scale spiral lobes in the near wake. The lobes define material transport pathways through which free stream fluid is entrained into the developing vortex structures, consistent with the strong mixing and organised Kármán shedding observed in the baseline flow. The alternating roll-up and interaction of the ridges are therefore consistent with the organised shedding process associated with the pronounced tonal response. At $\alpha =0.5$ (figure 15 b), the symmetry of the transport topology is broken. The ridge associated with the upper shear layer extends farther downstream before rolling up, whereas the lower-shear-layer ridge curls closer to the cylinder. This asymmetric organisation modifies the alternating entrainment pathways and is consistent with the dual-frequency response and reduced spanwise coherence discussed in § 3.1. At $\alpha =1$ (figure 15 c), rolled-up structures remain visible farther downstream, indicating that organised shedding persists, although the ridge pattern is less strongly entangled than in the baseline case. This behaviour is consistent with the partial re-establishment of periodic shedding identified from the pressure and velocity results. A marked change occurs at $\alpha =2$ (figure 15 d), for which the strongest noise reduction is obtained. The near-wake FTLE ridges become smoother, less convoluted and strongly deflected in the clockwise direction. Large-scale spiral lobes are substantially weakened compared with those at the lower rotation ratios. This change indicates a pronounced reduction in large-scale entrainment into the wake and weaker interaction between the upper and lower shear layers. The resulting material-transport topology is consistent with the comparatively weak and less organised wake, as well as with the reduced hydrodynamic fluctuations and acoustic radiation at this condition. At $\alpha =4$ (figure 15 e), the attracting material lines wrap closely around the cylinder in the direction of rotation, whereas the downstream wake contains fewer pronounced rolled-up structures than in the lower-rotation cases. This topology indicates that the large-scale entrainment and shear-layer interaction associated with classical Kármán shedding are strongly reduced. It does not, however, imply a completely steady flow, because localised 3-D disturbances remain within the near-wall rotating layer and cannot be fully represented by the present planar FTLE field.

Overall, the LCS analysis shows that rotation reorganises the in-plane material-transport pathways of the separated shear layers. The reduction in their mutual interaction, large-scale roll-up and entrainment into organised wake structures is consistent with the attenuation of the dominant shedding-related fluctuations.

3.4. Three-dimensional spatial correlation analysis

To visualise the 3-D spatial relationship between the surface-pressure and wake-velocity fluctuations and to identify wake regions statistically associated with the wall-pressure response, the zero-time-lag two-point cross-correlation coefficient field, $R_{pv}(\boldsymbol{x},\boldsymbol{x}_{{ref}})$ , is computed. Similar statistical approaches have been widely employed in simultaneous flow–pressure measurements to identify flow structures associated with surface-pressure events (Hudy et al. Reference Hudy, Naguib and Humphreys2007; Hernández Reference Hernández2022; Maryami & Ali Reference Maryami and Ali2023; Pan & Ye Reference Pan and Ye2025). The coefficient quantifies the linear correlation between the pressure fluctuation $p'(t)$ at a reference point $\boldsymbol{x}_{{ref}}$ on the cylinder surface and the transverse velocity fluctuation $v'(\boldsymbol{x},t)$ throughout the 3-D wake.

The cross-correlation coefficient is defined as (Pope Reference Pope2000)

(3.10) \begin{equation} R_{pv}(\boldsymbol{x},\boldsymbol{x}_{{ref}}) = \frac { \overline { p'(t,\boldsymbol{x}_{{ref}}) v'(t,\boldsymbol{x}) } }{ \sqrt { \overline { p'^2(t,\boldsymbol{x}_{{ref}}) } } \sqrt { \overline { v'^2(t,\boldsymbol{x}) } } }. \end{equation}

Here, the overbar denotes a time average, $\boldsymbol{x}_{{ref}}$ is the location of the pressure reference point on the cylinder surface, at either $\theta =90^\circ$ or $\theta =270^\circ$ , and $\boldsymbol{x}=(x,y,z)$ denotes a location within the wake. An isosurface of $R_{pv}$ identifies a spatial region in which the wake-velocity fluctuations are statistically correlated with the wall-pressure signal at zero time lag. A positive value indicates that pressure and transverse-velocity fluctuations tend to have the same sign, whereas a negative value indicates an anticorrelated relationship.

Figure 16 presents the 3-D isosurfaces of $R_{pv}$ for reference points on the upper ( $\theta =90^\circ$ , figure 16 i) and lower ( $\theta =270^\circ$ , figure 16 ii) cylinder surfaces. Because different visualisation thresholds are required for the different rotation conditions, the isosurfaces are compared primarily in terms of their morphology, orientation and spatial organisation rather than their absolute volume.

In the baseline case ( $\alpha =0$ , figure 16 a), the isosurfaces evaluated at $|R_{pv}|=0.60$ form large-scale, spanwise-organised structures that extend through the shear layers and into the near wake. Similar correlation patterns are obtained for the upper and lower reference points. This result indicates that the wall-pressure and wake-velocity fluctuations share the organised spatial signature of the classical alternating shedding process. At $\alpha =0.5$ (figure 16 b), the correlation topology becomes strongly asymmetric. Although the lower surface exhibits pronounced pressure fluctuations, as shown in figure 6(b), only weak zero-time-lag correlation with the sampled wake-velocity field is visible for the lower reference point, even at the reduced visualisation threshold. This contrast suggests that the locally enhanced pressure response is accompanied by reduced spatial organisation and/or altered phase relationships between the near-wall pressure and wake fluctuations. For the upper reference point, the correlation structures remain visible but become fragmented and less uniform along the span. Together with the reduced spanwise coherence, this loss of organised wall–wake correlation is consistent with the lower far-field radiation efficiency at this rotation ratio.

At $\alpha =1$ (figure 16 c), the correlation structures recover part of their spatial extent and organisation. The more regularly aligned patterns are consistent with the partial re-establishment of periodic shedding identified from the pressure and velocity results. Nevertheless, the correlation topology remains more asymmetric and less spanwise uniform than in the baseline case. At $\alpha =2$ (figure 16 d), the correlation field again exhibits pronounced asymmetry. For the lower reference point, the weak isosurface response indicates a substantial loss of zero-time-lag spatial correlation between the lower-surface pressure and the sampled wake-velocity fluctuations. For the upper reference point, the alternating wake structures observed in the baseline case are replaced by flatter, strip-like correlation regions. Their orientation is consistent with the strongly deflected and shortened upper shear layer and with the compression of the recirculation region under the Magnus effect. The resulting topology indicates that the residual pressure–velocity correlation is localised within the deflected near-wake flow rather than organised into large-scale alternating rollers.

Finally, at $\alpha =4$ (figure 16 e), the correlation topology differs markedly from that of the lower-rotation cases. For the upper reference point, streamwise strip-like structures remain close to the cylinder, indicating localised correlation within the near-wall rotating layer. For the lower reference point, a sheet-like correlation region wraps partially around the cylinder surface. This structure is consistent with the strong modification of the near-wall flow by rotation and with the suppression of the classical downstream shedding pattern. It should not, however, be interpreted as evidence of a completely steady, closed or fully attached flow layer, because localised 3-D disturbances remain present at this rotation ratio.

Figure 16.

Three-dimensional isosurfaces of the zero-time-lag cross-correlation coefficient $R_{pv}$ between the surface-pressure fluctuation $p'$ at a reference point and the transverse velocity fluctuation $v'$ in the wake. Subpanels (i) to (ii) correspond to pressure reference points on the upper ( $\theta =90^\circ$ ) and lower ( $\theta =270^\circ$ ) surfaces, respectively. Panels (a)–(e) correspond to the baseline case and rotation ratios $\alpha =0.5$ , $1$ , $2$ and $4$ , respectively. The visualisation levels are $|R_{pv}|=0.60$ for the baseline case; $0.20$ for $\alpha =0.5$ ; $0.45$ and $0.15$ for the upper and lower reference points at $\alpha =1$ , respectively; and $0.20$ for $\alpha =2$ and $4$ . Red and blue denote positive and negative correlations, respectively. Because different visualisation thresholds are used, the isosurfaces are compared primarily in terms of their morphology and spatial orientation rather than their absolute volume.

3.5. Dynamic mode decomposition

To examine the effects of rotational control on the spatial organisation and relative modal contributions of the wake and shear layers, DMD is applied to the instantaneous flow fields at different rotation ratios. Figure 17 shows the eigenvalue distributions associated with the first four dominant DMD modes in the complex plane. The horizontal and vertical axes represent the real and imaginary parts of the eigenvalues, respectively, and the black dashed circle denotes the unit circle ( $|\lambda |=1$ ). Within the DMD representation, modes lying on the unit circle are neutrally stable, whereas modes inside ( $|\lambda |\lt 1$ ) and outside ( $|\lambda |\gt 1$ ) the unit circle correspond to decaying and growing DMD modes, respectively. The first four dominant modes are colour-coded, while the remaining higher-order modes are shown in grey. In the baseline case (figure 17 a), the first mode (Mode 1, blue) resides on the unit circle, representing the stationary mean flow. Notably, the eigenvalue of Mode 2 lies outside the unit circle, indicating an unstable vortex-shedding DMD mode with a positive growth rate. This modal behaviour is consistent with the sustained periodic Kármán shedding observed in the flow and pressure spectra. At $\alpha = 0.5$ (figure 17 b), a stabilising effect is initially observed. The eigenvalues of all fluctuating modes, including Mode 2, retreat inside the unit circle. This indicates that the original Kármán-shedding DMD modes become stable or decaying under low-rate rotation. As rotation increases to $\alpha = 1$ (figure 17 c), a destabilising trend re-emerges. Both Mode 2 (green) and Mode 3 (pink) move outside the unit circle, signifying the excitation of multiple unstable modes. This implies the resurgence of the vortex shedding mode. As the rotation further increases to $\alpha = 2$ (figure 17 d), the eigenvalues of the dominant fluctuating modes (Modes 2–4) shift back inside the unit circle. This transition indicates substantial attenuation of the dominant shedding-related DMD modes, consistent with the weakened coherent vortex shedding and the significant far-field noise reduction. Finally, at $\alpha =4$ (figure 17 e), the dominant classical-shedding-related eigenvalues remain inside the unit circle, indicating stable or decaying DMD modes. This result confirms the attenuation of the classical Kármán mode, but does not imply a completely steady flow because localised low-frequency 3-D disturbances persist within the near-wall rotating layer.

Figure 17.

Distribution of eigenvalues in the complex plane under different rotation ratios: ( $ a$ ) baseline, ( $ b$ ) $\alpha = 0.5$ , ( $ c$ ) $\alpha = 1$ , ( $ d$ ) $\alpha = 2$ , ( $ e$ ) $\alpha = 4$ .

Figure 17. Long description

Panel A: A scatter plot showing the distribution of eigenvalues in the complex plane for the baseline case. The x-axis represents the real part of the eigenvalues, and the y-axis represents the imaginary part. The unit circle is marked, and different modes are highlighted with distinct colors. Panel B: A scatter plot similar to Panel A but for a rotation ratio of 1. The distribution of eigenvalues and modes is shown. Panel C: A scatter plot for a rotation ratio of 2, displaying the eigenvalues and modes. Panel D: A scatter plot for a rotation ratio of 3, showing the eigenvalues and modes. Panel E: A scatter plot for a rotation ratio of 4, illustrating the eigenvalues and modes. Each panel includes a zoomed-in inset to highlight details near the unit circle.

Figure 18.

Relationship between frequency and growth/decay rate under different rotation ratios: ( $ a$ ) baseline, ( $ b$ ) $\alpha = 0.5$ , ( $ c$ ) $\alpha = 1$ , ( $ d$ ) $\alpha = 2$ , ( $ e$ ) $\alpha = 4$ .

Figure 18 presents the modal growth/decay rate $\sigma$ as a function of frequency. Here, $\sigma \gt 0$ denotes a growing or unstable DMD mode, whereas $\sigma \lt 0$ denotes a decaying or stable DMD mode. The mean-flow mode remains approximately neutral, with $\sigma \approx 0$ . In the baseline case (figure 18 a), Mode 2 exhibits a positive growth rate and is associated with the sustained Kármán-shedding response. At $\alpha =0.5$ (figure 18 b), the growth rates of the dominant fluctuating modes become negative, indicating attenuation of the original shedding-related DMD modes. Growing modes re-emerge at $\alpha =1$ (figure 18 c), consistent with the partial recovery of organised shedding. At $\alpha =2$ (figure 18 d), all dominant fluctuating modes exhibit negative growth rates, corresponding to a strong stabilising modal trend and substantial attenuation of the shedding-related structures. At $\alpha =4$ (figure 18 e), the dominant classical-shedding-related modes remain decaying, although localised low-frequency near-wall disturbances are still present. Overall, the growing DMD modes observed at $\alpha =0$ and $1$ coincide with stronger tonal responses, whereas the negative growth rates of the dominant shedding-related modes accompany the substantial noise reduction obtained at $\alpha =0.5$ and $2$ .

Figure 19.

Spectral distribution of the relative DMD modal-energy indicator, $P_{\!j}$ , as a function of $St$ for the baseline case and different rotation ratios ( $\alpha =0.5$ , 1, 2 and 4).

To further elucidate the physical connection between the wake dynamics and the radiated noise, figure 19 presents the spectral distribution of the relative DMD modal-energy indicator, $P_{\!j}$ , as a function of $St$ . This indicator provides a quantitative measure of how rotation modifies the relative contributions of the dominant frequency-resolved flow structures. It is used for comparison between the DMD modes obtained using the same normalisation convention and is not interpreted as an absolute kinetic-energy spectrum. In the baseline case ( $\alpha =0$ ), the spectrum is dominated by a sharp peak at the primary vortex-shedding frequency, $St=0.19$ . This pronounced modal contribution is consistent with the strongly organised periodic vortex shedding observed in the flow field and with the prominent tonal peak in the acoustic spectrum. At $\alpha =0.5$ , two peaks appear at $St_{01}=0.18$ and $St_{02}=0.23$ , identifying two characteristic shedding-related components. Their magnitudes are lower than the dominant baseline contribution, consistent with the reduced tonal radiation shown in figure 2. At $\alpha =1$ , consistent with the re-emergence of growing DMD modes in the eigenvalue analysis, a pronounced modal-contribution peak reappears at the higher frequency $St\approx 0.24$ . This response is consistent with the partial re-establishment of organised periodic shedding, which limits the noise-reduction performance at this intermediate rotation ratio. Among the investigated cases, the strongest control effect is obtained at $\alpha =2$ . The relative modal-energy-indicator spectrum is substantially reduced over the analysed frequency range. The modal contribution near the classical shedding frequency, $St\approx 0.2$ , is strongly attenuated, while the higher-frequency contributions are also lower than those in the baseline case. These results indicate that the dominant shedding-related modes are strongly attenuated at $\alpha =2$ , with no pronounced modal concentration around the classical shedding frequency. For the high-rotation case ( $\alpha =4$ ), a broad modal contribution appears near $St\approx 0.1$ , exceeding the baseline level within this low-frequency range, whereas the classical shedding-frequency component remains weak. At higher frequencies, the velocity-based DMD modal contribution within the analysed flow volume remains relatively low compared with the broadband elevation observed in the acoustic spectrum. This difference reflects the distinct quantities represented by the two analyses: DMD characterises coherent velocity structures within the sampled flow volume, whereas the wall-integral FW–H formulation is directly sensitive to unsteady surface loading. Spatially localised near-wall disturbances may therefore produce appreciable broadband surface-pressure fluctuations while contributing relatively little to the velocity-based DMD modal-energy indicator.

To visualise the spatial organisation of the dominant flow scales identified from the characteristic peaks of the relative modal-energy-indicator spectra (figure 19), figure 20 presents the 3-D coherent structures of the key DMD modes. These structures are identified using isosurfaces of the $Q$ -criterion, which effectively isolates the vortex cores from the shear background. The surfaces are coloured by the spanwise vorticity ( $\omega _z$ ). For the baseline case ( $\alpha =0$ , figure 20 a), the mean flow mode ( $St=0$ ) exhibits a symmetric topology with two large-scale recirculation bubbles immediately behind the cylinder. The shear layers appear straight and extended before rolling up. The fluctuating mode at the fundamental frequency ( $St=0.19$ ) is visualised by robust $Q$ -criterion isosurfaces ( $Q=50\,000$ ), exhibiting the classic Kármán vortex street topology. The structures appear as large-scale, spanwise-coherent vortex tubes that alternate in sign and convect downstream. The strong spanwise coherence indicates a predominantly quasi-2-D organisation of the dominant shedding mode at this Reynolds number. The first harmonic mode ( $St=0.38$ ) displays similar spanwise coherence but with half the streamwise wavelength. At $\alpha =0.5$ (figure 20 b), the flow dynamics become more complex. The zero-frequency mode ( $St=0$ ) reveals the structural asymmetry induced by rotation: the upper shear layer, influenced by the rotation-induced momentum input, becomes elongated and comparatively smooth, whereas the lower shear layer exhibits distinct wavy structures. These wavy features spatially correspond to the region of elevated wall-pressure fluctuations identified in figure 6(b), indicating that the lower side remains dynamically active. For the fluctuating modes, consistent with the relative modal-energy-indicator spectrum, the $St=0.23$ mode exhibits the largest relative modal contribution. Crucially, as highlighted by the black dashed lines, the vortex crests of this mode display a pronounced spanwise phase dislocation. Unlike the spanwise-aligned shedding structures in the baseline case, this rotation-induced twisting provides a spatial interpretation of the reduced wall–wake coupling. Although the local modal contribution remains appreciable, as reflected by the wavy lower-shear-layer structure, the spanwise phase synchronisation is disrupted. This phase dislocation reduces the constructive addition of distributed loading fluctuations from different spanwise locations, thereby lowering the far-field radiation efficiency despite the locally enhanced pressure fluctuations. In contrast, the secondary mode at $St=0.18$ consists of fragmented and twisted structures confined to the near wake, indicating that the natural shedding tendency is spatially weakened.

Figure 20.

Three-dimensional spatial structures of the dominant DMD modes identified from characteristic peaks of the relative modal-energy-indicator spectra. The vortex structures are visualised using isosurfaces of the $Q$ -criterion and coloured by the normalised spanwise vorticity $\omega _z D/U_\infty$ (red for positive and blue for negative). All modes are visualised using $Q=50\,000$ , except for the baseline harmonic mode at $St=0.38$ , for which $Q=5000$ is used to reveal its weaker spatial structure. Panels (a)–(e) correspond to the baseline case and rotation ratios $\alpha =0.5$ , $1$ , $2$ and $4$ , respectively. The characteristic $St$ of each mode is indicated above the corresponding panel.

At $\alpha =1$ (figure 20 c), flow reorganisation is evident. The mean flow mode ( $St=0$ ) exhibits a topology similar to the baseline case, but with the recirculation region extending farther downstream. The dominant shedding mode shifts to a higher frequency ( $St=0.24$ ). Visually, this mode regains a degree of spanwise coherence, appearing as vertically aligned vortex sheets. This spatial reorganisation coincides with the recovery of the tonal peak in the modal-energy-indicator spectrum and is consistent with the partial re-establishment of organised periodic shedding under moderate rotation. A dramatic topological transition occurs at the optimal control ratio $\alpha =2$ (figure 20 d). The mean flow mode ( $St=0$ ) demonstrates a significant downward deflection of the wake, indicating a strong Magnus effect that suppresses the recirculation region. The fluctuating mode at $St=0.2$ becomes extremely weak. The $Q$ -criterion isosurfaces appear sparse and fragmented, failing to form organised large-scale rollers even at the same isovalue ( $Q=50\,000$ ) used for the baseline fundamental shedding mode. This lack of coherent vortical organisation is consistent with the strong attenuation of the dominant shedding-related modes.

Finally, at the high rotation ratio $\alpha =4$ (figure 20 e), the wake topology differs fundamentally from that of the lower-rotation cases. In the zero-frequency mode ( $St=0$ ), the near-wall structure on the lower side loses its spanwise uniformity and develops into streamwise-elongated structures extending into the downstream region. This spanwise discontinuity indicates that the flow no longer maintains a nominally 2-D attached configuration. Instead, the strong shear and centrifugal effects associated with rotation promote pronounced three-dimensionality, weakening the spanwise organisation of the near-wall rotating layer. The modes associated with the low-frequency contribution at $St=0.08$ and $0.12$ also differ markedly from the classical Kármán shedding modes. Rather than forming alternating spanwise-oriented rollers, the $Q$ -criterion isosurfaces reveal streamwise-elongated structures originating near the cylinder and extending downstream. Their spatial organisation is consistent with the low-frequency modulation of the near-wall rotating layer and with the Taylor–Görtler-like helical structures identified from the instantaneous vortex field. These modes therefore provide a plausible hydrodynamic link between the rotation-induced 3-D near-wall structures and the low-frequency pressure and acoustic response near $St\approx 0.1$ .