2.1. Wind tunnel and test section

The measurements were carried out in a low-speed, open-jet anechoic wind tunnel at the Southern University of Science and Technology (SUSTech) (Yang et al. Reference Yang, Liu, Liu, Shen, Zhang, Wei, Liu and Xu2021), as shown in figure 1(a). The closed-loop wind tunnel has a rectangular cross-section nozzle with a width of 600 mm and a height of 550 mm. The inner dimensions of the anechoic chamber are $3.8\,{\rm m} \times 5.7\,{\rm m} \times 3\,{\rm m}$ and the walls are covered with a broadband compact absorber with a thickness of 35 mm to achieve anechoic performance for frequencies greater than 100 Hz at a distance of 1.9 m from the centre of the test section. The wind tunnel is capable of producing flow speeds up to $70\,{\rm m}\,{\rm s}^{-1}$ with an incoming flow turbulence intensity of 0.15 %. The background noise level is 76 dBA at $70\,{\rm m}\,{\rm s}^{-1}$, which was measured 1.5 m away from the wind tunnel centreline. The test section involving parallel endplates, i.e. the top and bottom laser-cut plates attached to the nozzle exit, has a length of 800 mm and the alignment of the rectangular endplates helps to maintain a nearly 2-D flow over the cylinder. According to the coordinate system with an origin located at the midspan of the cylinder, the nozzle exit is located at $x=-300$ mm. Therefore, the cylinder was properly placed within the potential core of the nozzle jet flow.

Figure 1. Schematic diagram depicting (a) the experimental set-up with a cylinder in the aeroacoustic wind tunnel and (b) the top view of the planar PIV configuration.

2.2. Cylinder design

A cylinder model with an outer diameter of $D=64.7$ mm and a span of $L=550$ mm was designed. The cylinder was composed of three different parts along its span, consisting of one middle section with pressure taps and chamber instrumentation and two side extension parts, as depicted in figure 2(a). Several factors were considered to determine the suitable value of the outer diameter of the cylinder. The limiting factors were the blockage ratio, aspect ratio and the effects of low-frequency background noise. By considering the outer diameter of 64.7 mm, the blockage was found to be less than 12 %. The cylinder had an aspect ratio of approximately 8.5, which indicates that any influence from the endplates can be ignored (Zdravkovich Reference Zdravkovich1981).

Figure 2. (a) Schematic of the cylinder design, (b) the $xy$-plane of the cylinder and (c) the sensing area on the cylinder equipped with pressure taps, ${P}_{i}$ ($i=1:20$), distributed around the circumference of the cylinder at midspan. Note that only pressure taps $P_{6}$ to $P_{14}$ are visible in this diagram. The peripheral locations of all 20 pressure taps are listed in table 2.

By considering a balance between low background noise and high vortex-shedding frequency, the incoming flow velocity was set at $U_{\infty }=20\, {\rm m}\,{\rm s}^{-1}$, corresponding to $Re=7\times 10^{4}$. To assist the explanation of the cylinder design process, the $St=f\kern0.7pt D/U_{\infty }$ relationship is provided, where $f$ (Hz) is vortex shedding tonal frequency. If it is assumed that $St\approx 0.19$ for the anticipated Reynolds number range of a circular cylinder (Norberg Reference Norberg2003) then $f\approx 59$ Hz at $U_{\infty }=20\, {\rm m}\,{\rm s}^{-1}$ should be observed. From a previous experiment (Maryami et al. Reference Maryami, Arcondoulis, Liu and Liu2023c) conducted in the same wind tunnel at SUSTech in a flow speed range of $U_{\infty }=15\unicode{x2013}50\,{\rm m}\,{\rm s}^{-1}$, vortex shedding tones from the cylinder at this frequency typically protrude above the background noise by at least 10 dB. Note that the Reynolds number considered here is matched to the subcritical regime (Roshko Reference Roshko1955; Achenbach Reference Achenbach1968), which is the flow conditions of typical aeroacoustic applications as stated in § 1. In addition, the PIV measurements at $U>30\, {\rm m}\,{\rm s}^{-1}$ are not recommended due to difficulties in stabilising the flow field. Therefore, the specific flow velocity of $U_{\infty }=20\, {\rm m}\,{\rm s}^{-1}$ was strategically selected to achieve an optimal particle seeding density, an accurate and reliable tracking of the particles within the fluid, and hence robust data of flow velocity.

The middle section of the cylinder was coated with a series of solid layers (cylinders with inner and outer diameters of $d=40$ mm and $D=64.7$ mm, respectively, see figure 2b) containing holes with a diameter of $d_{0}=1.2$ mm to allow distributed LB to be applied through the surface of the cylinder as shown in figure 2(c). In this experimental study, LB was implemented at circumferential angles of $\theta _{b}={\pm }41^{\circ }$, ${\pm }90^{\circ }$ and ${\pm }131^{\circ }$ in an open-loop control configuration. These cases are denoted as LB41, LB90 and LB131, respectively, representing the application of LB in the boundary layer, shear layers on the cylinder, and separated shear layers, respectively. Hence, the LB holes with a spanwise spacing of $L_{0}=7$ mm were distributed along two lines: one positioned on the top half and the other on the bottom half of the cylinder. The experimental data provided in the literature show that the spanwise correlation length of a circular cylinder extends to approximately $3.3D$ at $Re\approx 7\times 10^{4}$ (Norberg Reference Norberg2003), yet there is a lack of fidelity experimental or numerical data to determine this length in the case of LB at a specific Reynolds number and air flow rate. Therefore, a total of 64 blowing holes were evenly distributed along each line, covering a span length of $7D$, to ensure the development of a sufficient 3-D flow field along the cylinder span.

2.3. Chamber design

In the present study, the air supply system was composed of three main components: an air supply device (compressor), air preparation equipment (air filter, regulator, pressure gauge and flow meter) and an air flow chamber. All these components except the airflow chamber were located outside the anechoic chamber. Three 3-D printed chambers were designed similar to Maryami et al. (Reference Maryami, Arcondoulis, Yang, Szoke, Xiang, Guo, Wei and Liu2022b); Maryami, Arcondoulis & Liu (Reference Maryami, Arcondoulis and Liu2023a,Reference Maryami, Arcondoulis and Liub). Each chamber was installed firmly in the corresponding solid layer as shown in figure 2(b), so that the blowing holes of the chamber were aligned properly with those of the solid layer. The chamber was further divided at the half-span of the cylinder to allow for remote-sensing pressure taps to be embedded at the midspan. Therefore, LB was applied to both sides of the chamber, where each side was fed by eight inlets for tube connections (see figure 2a). The design of a series of chambers ensured a more uniform distribution of LB along the span. To check the uniformity of LB along the span, the static pressure distribution was measured through a total of 32 steel tubes with inner and outer diameters of 0.5 and 1 mm, respectively, inserted into the blowing holes of the test chamber. The steel tubes were connected to two 16-channel Model 9116 intelligent pressure scanners with a full-scale pressure range from 2.5 kPa to 5200 kPa by flexible tubes with an inner diameter of 0.5 mm and an outer diameter of 2 mm. The results showed that the pressure along the chamber span is nearly constant within ${\pm }0.1\,\%$ as presented in figure 3.

Figure 3. Static pressure, $p_{st}$, distribution along the half-span of the chamber normalised by the mean static pressure, $\bar {p}_{st}$, in the case of LB90.

The LB flow rate was measured by a mass flow meter and was kept constant by using a regulator and pressure gauge. The equivalent momentum coefficient, $C_{\mu }$, which is the ratio of LB momentum flux at each hole to incoming flow momentum flux, was used to facilitate the research (Lin et al. Reference Lin, Towfighi and Rockwell1995; Chen, Li & Hu Reference Chen, Li and Hu2014; Gao et al. Reference Gao, Chen, Chen, Huang and Li2019; Deng et al. Reference Deng, Gao, Chen and Chen2021; Maryami et al. Reference Maryami, Arcondoulis, Yang, Szoke, Xiang, Guo, Wei and Liu2022b, Reference Maryami, Arcondoulis and Liu2023a,Reference Maryami, Arcondoulis and Liub) and is defined as follows:

(2.1)\begin{equation} C_{\mu}=\left(\frac{U_{b}}{U_{\infty}}\right)^{2}\times\left(\frac{A_{b}}{DL_{0}}\right), \end{equation}

where $U_{b}$ (${\rm m}\,{\rm s}^{-1}$) is the average LB velocity of each hole, $L_{0}$ (m) is the spanwise spacing between two neighbouring LB holes, and $A_{b}$ ($\mathrm {m}^{2}$) refers to the combined area of the two blowing holes, one located on the top side and the other on the bottom side of the cylinder. The experiments were also carried out for three air flow rates of $Q_{b}=12$, 20 and $28\,\mathrm {m}^{3}\,\mathrm {h}^{-1}$. The $Q_{b}$, $U_{b}/U_{\infty }$, and corresponding $C_{\mu }$ values of LB cases are presented in table 1. Please note that significant tonal noise reduction was achieved for all $C_{\mu }$ values within this $Q_{b}$ range. However, in this study, only the case with $C_{\mu }\approx 0.007$ ($Q_{b}=12\, \mathrm {m}^{3}\,\mathrm {h}^{-1}$) was specifically presented as it serves as an indicative example for all the investigated cases.

Table 1. Blowing flow rate, $Q_{b}$, velocity ratio, $U_{b}/U_{\infty }$ and corresponding $C_{\mu }$ values calculated using (2.1).

2.4. Near-field pressure and acoustic measurements

Near-field pressure measurements were performed using a remote-sensing technique and pressure taps installed in the middle section of the cylinder. The cylinder was equipped with 20 pressure taps distributed with non-uniform angular spacing as shown in figure 2(c) and their peripheral locations are summarised in table 2. The pressure taps are comprised of a brass tube with inner and outer diameters of 0.8 mm and 1.6 mm, respectively, fitted in a designated housing on the solid-layer surface. As shown in figure 4(a), the brass tubes were connected to the remote-sensing probes using brass and flexible polyurethane tubing with inner and outer diameters of 0.8 and 4 mm, respectively, similar to that used by Maryami et al. (Reference Maryami, Arcondoulis, Liu and Liu2023c). Each remote probe consists of a microphone holder and a Panasonic WM-61A microphone with a circular sensing diameter of 2 mm attached to a pinhole with a diameter of 0.4 mm (see figure 4b). An extension tube of 2 m was connected to the microphone-holder brass tube to eliminate the spurious tones related to the standing waves in the flexible tubing. Panasonic WM-61A microphones have been used by many (Angland et al. Reference Angland, Zhang and Goodyer2012; Maryami et al. Reference Maryami, Azarpeyvand, Dehghan and Afshari2019, Reference Maryami, Showkat Ali, Azarpeyvand and Afshari2020; Vemuri et al. Reference Vemuri, Liu, Zang and Azarpeyvand2020; Maryami et al. Reference Maryami, Ali, Azarpeyvand, Dehghan and Afshari2022a, Reference Maryami, Arcondoulis, Liu and Liu2023c,Reference Maryami, Arcondoulis, Yang and Liud) to produce reliable pressure data over the frequency range of interest $(20\,{\rm Hz} \leq f \leq 10 \,{\rm kHz})$.

Figure 4. Schematic of the remote-sensing set-up for (a) the full view of the cylinder and (b) assembly view of the remote-sensing holder.

Table 2. Peripheral positions of the pressure taps, ${P}_{i}$ ($i=1:20$), at $z/D=0$.

In situ calibration was carried out for the remote-sensing probes to compensate for any possible amplitude damping within the small tubes and the lag produced by the remote probe extensions. A tube with a length of 110 mm and a diameter of 10 mm along with a cone connected to a high-quality loudspeaker was used for the calibration similar to Maryami et al. (Reference Maryami, Azarpeyvand, Dehghan and Afshari2019). The reference microphone was a 1/2 inch Brüel & Kjær (B&K) free-field microphone Type 4966 that was calibrated using a B&K calibrator Type 4231. A transfer function between the remote and reference microphones was obtained using a white noise signal, following the procedure of Mish (Reference Mish2001). The coherence values between these pressure signals were smooth and near one for frequencies less than or equal to approximately 6 kHz. Note that this frequency corresponds to $St\approx 19.5$ at $U_\infty =20\, {\rm m}\,{\rm s}^{-1}$, which is sufficiently high for the frequency range of interest in this experimental investigation and is similar to the upper-frequency range of others (Garcia-Sagrado & Hynes Reference Garcia-Sagrado and Hynes2011; Gruber Reference Gruber2012).

The far-field noise was measured using a B$\&$K free-field Type 4966 microphone located perpendicular to the flow and a distance of 1.5 m away from the cylinder axis as shown in figure 1. A National Instruments (NI) PXI-10420 chassis with two 24-bit synchronised NI PXI-4496 data acquisition (DAQ) cards were used to measure near-field pressure and far-field noise simultaneously at a sampling frequency of 51.2 kHz over 10 s. To convert the pressure–time data into the frequency domain, a fast Fourier transformation with Hamming windowing with 50 % overlap was used based on Welch's method, resulting in a frequency resolution of 2 Hz. These data were then converted into a power spectral density (PSD), namely $\phi _{pp,s}$ and $\phi _{pp}$ (dB Hz$^{-1}$) for near-field pressure and far-field noise, respectively, using a reference pressure of $p_{_0}=20\,\mathrm {\mu }{\rm Pa}$.

2.5. The PIV set-up

Time-resolved planar PIV experiments were conducted in the leeward region of the cylinder and in the near wake, where the vortex shedding occurs and is generally accepted to constitute the sound source. The field of view (FOV) (approximately $32\times 19\,\mathrm {cm}^{2}$) covers the flow field with dimensions of $5D\times 3D$ in the $x$- and $y$-directions, respectively, as shown in figure 1(b). The air in the tunnel was seeded with dioctyl sebacate particles with a mean diameter of $1\,\mathrm {\mu }{\rm m}$ and illumination was provided by a Photonics Nd: YAG laser (50 mJ $@$ 10 kHz). The light sheet was directed from the wake towards the cylinder, facilitating particle illumination on both the top and bottom sides of the centreline. A Phantom VEO410L ($1280 \times 780$ pixels, 12 bit, pixel pitch $20\,\mathrm {\mu }{\rm m}$, maximum frame rate 5.2 kHz) equipped with a Nikon Micro-Nikkor 60 mm objective was used at a focal ratio of $f/2.8$. Synchronisation of the laser pulses and image acquisition was accomplished using a LaVision programmable timing unit in combination with the LaVision DaVis 10.0 software package. Image pairs were acquired at a sampling frequency of 2.645 kHz over a duration of 2 s. All 5289 images were processed using an iterative multipass technique with a final window size of $32\times 32$ pixels and an overlap factor of 75 %. To address the issue of surface reflections in the PIV process the cylinder surface was painted black and the angle of incidence of the laser light was adjusted to optimise the illumination of the flow. By carefully selecting this angle, the amount of light that was directly reflected into the camera was minimised, resulting in an improved image quality for PIV analysis.

The transistor–transistor logic signal from the Q-switch of the first laser was recorded simultaneously with the microphone signal to synchronise the PIV with the microphone measurements. Henning et al. (Reference Henning, Kaepernick, Ehrenfried, Koop and Dillmann2008) reported that the leading edge of the transistor–transistor logic signal corresponds to the firing of the laser pulse and hence provides the time of the captured velocity field snapshot. The Q-switch signal was acquired with the same DAQ system as the microphone signal (NI PXI-4496), also sampled with a rate of 51.2 kHz. The NI PXI-4496 DAQ was also triggered by the programmable timing unit to ensure synchronous measurements of the PIV system and pressure data. As the PIV and pressure signals have different lengths and sampling rates, an antialiasing (low-pass) finite impulse response filter was used to resample the signals with the lower sampling rate during the resampling process. To compensate for any delay, the signals were aligned by the earliest signal.

3.1. Far-field noise spectra

Far-field acoustic PSD, $\varPhi _{pp}$ (dB Hz$^{-1}$), as a function of $St$ for each cylinder is presented in figure 5 and is calculated from

Figure 5. Acoustic PSD, $\varPhi _{pp}$, plotted against $St$ for various LB cases. The labels $f_{1}$, $f_{2}$ and $f_{3}$ representthe fundamental vortex shedding tone (first harmonic), second and third harmonics, respectively.

(3.1)\begin{equation} \varPhi_{pp}=10\log_{10}\left(\frac{\phi_{pp}}{p^{2}_{0}}\right). \end{equation}

The results show the existence of a tonal peak at $St\approx 0.19$ for the baseline, which corresponds to the fundamental vortex shedding tone ( $f_{1}$). This tonal peak falls within the published Strouhal number envelope for the tested Reynolds number according to Norberg (Reference Norberg2003). The $f_{1}$-tone is also called the first harmonic, other harmonics with frequencies that are positive integer multiples of the fundamental frequency are known as higher harmonics ( $f_{2}=2f_{1}, f_{3}=3f_{1}, f_{4}=4f_{1},\ldots$). The results reveal that every LB case except LB90 is able to reduce the $f_{1}$-tone relative to the baseline. It is observed that LB41 possesses superior performance in terms of tonal noise reduction of 19 dB, yet presents an increase in broadband energy content beyond $St\approx 1$. The LB41 also shows the emergence of another vortex shedding peak at $St\approx 0.25$, indicating the existence of two types of vortical structures in the wake region. For LB90, an increase in both tonal and broadband energy contents is observed, indicating that the application of LB perpendicular to the free stream near the boundary layer separation point, which is around $73.6^{\circ }$ for the baseline as reported by Maryami et al. (Reference Maryami, Arcondoulis, Liu and Liu2023c), promotes vortex shedding and is unfavourable to aerodynamic noise suppression. It is observed that LB131 reduces the $f_{1}$-tone by around 12 dB and reduces the broadband energy content within 8–10 dB up to $St\approx 2$. Note that the $f_{1}$-tone in the case of LB131 protrudes by approximately 23 dB above the broadband energy content, while this protrusion is of the order of 10 dB for LB41. It is observed that beyond $St\approx 2$, specifically within $St\approx 7\unicode{x2013}9$, broadband peaks are evident for all LB cases, which can potentially be attributed to the blowing self-noise phenomenon, as documented by Angland et al. (Reference Angland, Zhang and Goodyer2012). It is difficult to determine which part of the blowing system (i.e. compressor, chamber or air jets) this noise may be linked to. Nonetheless, the ratio of vortex shedding noise to blowing self-noise is sufficiently high as reported by Guo et al. (Reference Guo, Maryami, Yang, Yang, Wang and Liu2023) using the same air supply system at SUSTech and similar air flow rates. This finding is important in facilitating the study of the effects of LB on suppressing vortex shedding noise from a circular cylinder.

By integrating the far-field PSD over the whole frequency range of interest corresponding to $St\approx 0.13\unicode{x2013}13$, the overall sound pressure level (OASPL) is calculated as follows:

(3.2)\begin{equation} \mathrm{OASPL}=10\log_{10}\left(\frac{\int\phi_{pp}\,\mathrm{d}f}{p^{2}_{0}}\right), \end{equation}

and it is observed that the OASPL in the case of LB131 is lower in magnitude than that of LB41 by around 2.2 dB. Considering that the OASPL difference between LB41 and LB131 is not significant and the tonal noise is much more important in the case of bluff bodies, it can be argued that LB41 is still the better blowing method in terms of noise reduction.

3.2. Near-field pressure spectra

To gain further insights into the noise generation mechanism of the cylinder, the near-field pressure PSD, $\varPhi _{pp,s}$ (dB Hz$^{-1}$), was calculated by (3.1) and the results at some angles, i.e. $\theta =0^{\circ }$, $41^{\circ }$, $90^{\circ }$, $131^{\circ }$ and $180^{\circ }$, are presented in figure 6. The $\varPhi _{pp,s}$ spectra of the baseline show a dominant tonal peak at the $f_{1}$-tone frequency, consistent with the fundamental tone observed in figure 5. Therefore, the surface pressure fluctuations are clearly linked to vortex shedding and are responsible for sound source generation (Inoue & Hatakeyama Reference Inoue and Hatakeyama2002). Although the concept of the feedback signal (upstream influence) in the context of vortex shedding from a circular cylinder has already been observed in the experiments by Nishioka & Sato (Reference Nishioka and Sato1978) and Unal & Rockwell (Reference Unal and Rockwell1988); the precise mechanism by which vortex shedding affects the cylinder remains unclear. It is uncertain whether this effect is caused by the induced velocity field of the vortices or by upstream-travelling vorticity waves that originate from the initial global instability (Williams et al. Reference Williams, Mansy and Amato1992). In both cases, however, the vorticity dynamics in the wake modulates the boundary layer by providing initial spatiotemporal disturbances to the shear layers on the cylinder. These disturbances are then spatially amplified to a saturation level in the separated shear layers, leading to the formation of fully developed vortices in the wake region. These vortices produce a feedback signal to the cylinder, which influences the boundary layer and starts the cycle again.

Figure 6. Near-field PSD, $\varPhi _{pp,s}$, at $\theta =0^{\circ }$, $41^{\!\circ }$, $90^{\circ }$, $131^{\!\circ }$ and $180^{\circ }$. The labels $f_{1}$, $f_{2}$ and $f_{3}$ represent the fundamental vortex shedding tone (first harmonic), second and third harmonics, respectively.

The influence of vortex shedding on various peripheral locations around the cylinder, known as the feedback signal, is dependent on the distance between each location and the vortex formation region. As this distance increases, the strength of the feedback signal diminishes due to enhanced energy dissipation. This is further supported by the observations depicted in figure 6. It is observed that the $f_{1}$-tone (associated with lift fluctuations) with a magnitude of around 80 dB appears at $\theta =0^{\circ }$ and follows a gradually increasing trend with $\theta$ to reach a maximum of approximately 123 dB at $\theta =90^{\circ }$. This is because the distance between these measurement points and the vortex-formation region decreases, allowing for stronger interactions between the shed vortices and the nearby locations through feedback signal. For $\theta >90^{\circ }$ while the distance to the vortex-formation region continues to decrease with increasing $\theta$, the magnitude of the $f_{1}$-tone decreases until it becomes fully broadband at $\theta =180^{\circ }$. This difference can be attributed to the distinct roles played by surface pressure fluctuations on the leeward side of the cylinder compared with the windward side. Instead of contributing primarily to lift fluctuations, the surface pressure fluctuations on the leeward side of the cylinder have a greater impact on drag fluctuations, as will be depicted in figure 7. This shift in the characteristics of surface pressure fluctuations is evident in the magnitude of the $f_{2}$-tone, which increases as $\theta$ approaches $180^{\circ }$ (Casalino & Jacob Reference Casalino and Jacob2003; Maryami et al. Reference Maryami, Azarpeyvand, Dehghan and Afshari2019, Reference Maryami, Showkat Ali, Azarpeyvand and Afshari2020, Reference Maryami, Ali, Azarpeyvand, Dehghan and Afshari2022a, Reference Maryami, Arcondoulis, Liu and Liu2023c; Maryami & Ali Reference Maryami and Ali2023).

Figure 7. (a) Lift coefficient PSD, $\phi _{_{C_{_L}C_{_L}}}$, and (b) drag coefficient PSD, $\phi _{_{C_{_D}C_{_D}}}$, as a function of $St$. The labels $f_{1}$ and $f_{3}$ represent the odd harmonics, while $f_{2}$ and $f_{4}$ represent the even harmonics.

The variations in the size of the vortex-formation region influenced by local blowing cases have notable effects on the feedback signal and subsequently the near-field pressure characteristics at the $f_{1}$- and $f_{2}$-tone frequencies. In the cases of LB41 and LB131, where the vortex formation region is larger compared with the baseline (refer to § 4), the near-field pressure at these frequencies becomes weaker in terms of energy content yet LB90 exhibits the opposite behaviour. This conclusion is generally valid, except in the case of LB131 at the stagnation point. Despite having a larger vortex-formation region similar to LB41, LB131 exhibits an increase of 11 dB in the magnitude of the $f_{1}$-tone from the baseline at the stagnation point. On the other hand, LB41 demonstrates a broadband behaviour at the $f_{1}$-tone frequency, suggesting a distinct vortex-shedding mechanism for LB131 compared with LB41 (refer to § 5.4). At $\theta =0^{\circ }$, LB90 reveals the $f_{1}$-tone with a magnitude of approximately 6 dB. In addition to the $f_{1}$-tone, the first two harmonics, i.e. the $f_{2}$- and $f_{3}$-tones, are observed for LB90 in which the even harmonic is stronger than the odd harmonic, indicating that LB90 increases drag fluctuations. It will be shown in figure 7 that lift and drag fluctuations occur at the odd ( $f_{1}, f_{3},\ldots$) and even ( $f_{2}, f_{4},\ldots$) harmonics, respectively, as documented by many (Etkin et al. Reference Etkin, Korbacher and Keefe1957; Casalino & Jacob Reference Casalino and Jacob2003; Qin Reference Qin2004; Maryami et al. Reference Maryami, Showkat Ali, Azarpeyvand and Afshari2020, Reference Maryami, Ali, Azarpeyvand, Dehghan and Afshari2022a, Reference Maryami, Arcondoulis, Liu and Liu2023c; Maryami & Ali Reference Maryami and Ali2023).

Away from the stagnation point at $\theta =41^{\circ }$, $90^{\circ }$ and $131^{\circ }$ (see figure 6b–d), the $f_{1}$-tone appears for all LB cases and increases in magnitude with $\theta$, which is due to the shorter distance between these peripheral locations and the vortex-formation region. While LB41 shows the tonal peak of dual vortex shedding around the $f_{1}$-tone frequency as already observed in figure 5, it exhibits a remarkable reduction in the magnitude of the $f_{1}$-tone from the baseline, with a range of 23–30 dB, making it the most effective LB case. In contrast, LB90 demonstrates an increase in the magnitude of the $f_{1}$-tone compared with the baseline, reaching approximately 11 dB at $\theta =131^{\circ }$, indicating the presence of significant lift fluctuations. In the base region at $\theta =180^{\circ }$, as shown in figure 6(e), LB41 shows a broadband behaviour over the frequency range yet LB90 and LB131 present tonal peaks at the $f_{2}$- and $f_{1}$-tone frequencies, respectively, similar to the observations at $\theta =0^{\circ }$.

It is observed up to $\theta =90^{\circ }$ as shown in figure 6(a–c), that each LB displays a high broadband contribution at $St\gtrapprox 1$. For $\theta >90^{\circ }$, only LB90 increases the broadband energy content from that of the baseline. It can be argued that small-scale high-frequency fluctuations in the boundary layer that have moved up to the separation point contribute to the broadband components of the surface pressure. At $\theta =41^{\circ }$, it is notable that these structures appear to be independent of the size of the vortex-formation region. Moreover, any external perturbation applied to the boundary layer (LB41), shear layer on the cylinder (LB90) or separated regions (LB131) does not contribute to an increase in their energy content. Beyond separation, the high-frequency broadband pressure spectrum is attributed to small-scale high-frequency fluctuations of turbulence in the near-wake region (Doolan Reference Doolan2009), whose energy content decreases with increasing vortex formation size.

3.3. Lift and drag spectra

In order to study how LB cases affect the unsteady forces, figure 7 presents the spectra of the lift and drag coefficients, i.e. $\phi _{C_{_L}C_{_L}}$ and $\phi _{C_{_D}C_{_D}}$ (Hz$^{-1}$), respectively. The lift and drag spectra were calculated by integrating the surface pressure fluctuations around the circumference of the cylinder as done by others (Fage & Falkner Reference Fage and Falkner1931; Sadeh & Saharon Reference Sadeh and Saharon1982; Casalino & Jacob Reference Casalino and Jacob2003; Norberg Reference Norberg2003; Maryami et al. Reference Maryami, Showkat Ali, Azarpeyvand and Afshari2020, Reference Maryami, Ali, Azarpeyvand, Dehghan and Afshari2022a, Reference Maryami, Arcondoulis, Liu and Liu2023c; Maryami & Ali Reference Maryami and Ali2023). Note that the contribution of the skin friction was neglected because it is less than 2 % of the total drag in the subcritical Reynolds number range (Fage & Falkner Reference Fage and Falkner1931; Achenbach Reference Achenbach1968; Sadeh & Saharon Reference Sadeh and Saharon1982).

The results in figure 7 reveal dominant tonal peaks at the $f_{1}$- and $f_{2}$-tone frequencies in the $\phi _{C_{_L}C_{_L}}$ and $\phi _{C_{_D}C_{_D}}$ spectra, respectively, for the baseline, which is consistent with others (Casalino & Jacob Reference Casalino and Jacob2003; Qin Reference Qin2004; Maryami et al. Reference Maryami, Showkat Ali, Azarpeyvand and Afshari2020, Reference Maryami, Ali, Azarpeyvand, Dehghan and Afshari2022a, Reference Maryami, Arcondoulis, Liu and Liu2023c; Maryami & Ali Reference Maryami and Ali2023). The dominant tone frequency of $\phi _{C_{_D}C_{_D}}$ is twice that of $\phi _{C_{_L}C_{_L}}$, and thus the amplitude of fluctuating drag, which is significantly smaller than that of fluctuating lift (Bouak & Lemay Reference Bouak and Lemay1998; Posdziech & Grundmann Reference Posdziech and Grundmann2007), is dominated by fluctuating pressures that are in phase between the upper and lower sides of the cylinder. Interestingly, the dominant tonal peaks in the spectra of $\phi _{C_{_L}C_{_L}}$ and $\phi _{C_{_D}C_{_D}}$ are similar to those of the near-field spectra at $\theta =90^{\circ }$ and $180^{\circ }$, respectively, (compare figures 6c and 6e with figures 7a and 7b, respectively). These observations are consistent with previous findings (Casalino & Jacob Reference Casalino and Jacob2003; Maryami et al. Reference Maryami, Showkat Ali, Azarpeyvand and Afshari2020, Reference Maryami, Ali, Azarpeyvand, Dehghan and Afshari2022a, Reference Maryami, Arcondoulis, Liu and Liu2023c; Maryami & Ali Reference Maryami and Ali2023) and demonstrate that the alternating pressure fluctuations at the shoulders of the cylinder are lift fluctuations sources (Gerrard Reference Gerrard1966), whose energy is concentrated to a band around the fundamental vortex-shedding frequency (Gerrard Reference Gerrard1961; Ferguson & Parkinson Reference Ferguson and Parkinson1967). The dominant sound correlates with the fundamental frequency of lift fluctuations in a dipole pattern with a dipole axis normal to the free stream (Gerrard Reference Gerrard1955; Etkin, Korbacher & Keefe Reference Etkin, Korbacher and Keefe1956).

Each LB case exhibits tonal peak characteristics similar to the baseline. Specifically, LB90 demonstrates odd harmonics ( $f_{1}$ and $f_{3}$) and even harmonics ( $f_{2}$ and $f_{4}$) in the $\phi _{C_{_L}C_{_L}}$ and $\phi _{C_{_D}C_{_D}}$ spectra, respectively, with magnitudes exceeding those of the baseline. The significant fluctuations in lift and drag are attributed to the presence of strong vortex shedding, characterised by closely rolling up shear layers near the cylinder and exhibiting a notable spanwise correlation (Norberg Reference Norberg2003). In contrast, LB41 shows the superior reduction; the double $f_{1}$-tone in the $\phi _{C_{_L}C_{_L}}$ spectrum in line with the near- and far-field pressures (see figures 5 and 6, respectively), is reduced by at least two orders of magnitude relative to the baseline. In addition, the $\phi _{C_{_D}C_{_D}}$ spectrum is quite broadband over the frequency range.

4.1. Time-averaged flow fields

Contour plots of the normalised time-averaged $x$- and $y$-components of velocity, $\bar {u}/U_{\infty }$ and $\bar {v}/U_{\infty }$, respectively, are presented in figure 8. The baseline shows the emergence of a pair of counter-rotating vortices close to the base of the cylinder with an identical symmetric vortex core structure with respect to the centreline ($\kern0.7pt y/D=0$). Vortex structures are evident in each LB case, indicating that none of the LB techniques investigated in this study are able to completely suppress the generation mechanism of vortex shedding. Despite this, some LB cases are more capable of expanding or shifting the recirculation region downstream in comparison with the baseline. The observations indicate that LB41 enables the shear layers to extend farther downstream before rolling up, resulting in a larger vortex-formation region. In contrast, LB90 reduces the size of the vortex-formation region compared with the baseline, suggesting an acceleration in the rolling up of the shear layers into the wake. The time-averaged flow fields along with the observations in figures 5–7 reveal that a delay in vortex shedding induces a weak near-field pressure with low-energy lift oscillations at the $f_{1}$-tone frequency and then a superior tonal noise reduction. A similar observation was made by Sueki et al. (Reference Sueki, Takaishi, Ikeda and Arai2010) and Showkat Ali, Mahdi & Da Silva (Reference Showkat Ali, Mahdi and Da Silva2020) who demonstrated that a delay in flow recirculation using porous media as a form of passive flow control gives rise to more stable shear layers (relative to a baseline case) and noise reduction.

Figure 8. Normalised time-averaged $x$-and $y$-components of velocity, (a–d) $\bar {u}/U_{\infty }$ and (e–h) $\bar {v}/U_{\infty }$, respectively, superimposed by the time-averaged streamline. The solid red circle, square and triangle symbols represent the vortex core centre at $x\approx L_{vc}$, the end of the recirculation region at $x\approx L_{r}$ (saddle point) and the end of the vortex-formation region at $x\approx L_{f}$, respectively. The white contour region in $\bar {u}/U_{\infty }$ shows the flow acceleration area (FAA).

Figure 8 shows that in the cases of LB41 and LB90, the foreheads of the recirculation bubbles are observed at $x/D\approx 0.5$, whereas LB131 displays its forehead close to $x/D\approx 0.7$. This observation suggests that the vortex-shedding mechanism is primarily affected by the blowing applied on the windward and leeward sides of the cylinder and leads to a particular flow behaviour in the recirculation region. Furthermore, the forehead position of the recirculation bubble may help explain the mechanism of the increase in broadband spectral content in the cases of LB41 and LB90 compared with the baseline (see figure 5). According to Doolan (Reference Doolan2009), the near-wake region of a circular cylinder is characterised by turbulent flow exhibiting 3-D velocity fluctuations across different frequencies and phases. These turbulent fluctuations directly influence the pressure fluctuations on the cylinder surface, contributing to its acoustic signature. In addition, the presence of a high-frequency broadband spectrum in both near-field pressure and far-field noise can be attributed to small-scale, high-frequency turbulence fluctuations occurring in the near-wake region. Hence, the attachment of the recirculation bubbles forehead to the rear side of the cylinder in the cases of LB41 and LB90 is expected to generate small-scale high-frequency fluctuations with higher-energy content in the near-wake region. This phenomenon contributes to the observed high broadband spectral content in figure 5.

The emergence of a FAA is observed in figure 8(a–d), which can be related to the development of the recirculation region and the minimum pressure imposed in the vortex core. In particular, the depression region reaches its minimum value specifically in the zones where the time-averaged footprints of the von Kármán vortices are located (Greco et al. Reference Greco, Paolillo, Astarita and Cardone2020). Showkat Ali, Mahdi & Da Silva (Reference Showkat Ali, Mahdi and Da Silva2018) emphasised that the pressure difference between the inside and outside of the vortex core creates a partial vacuum (suction), which facilitates the acceleration of flow in the near-wake region.

In figure 8(a), the FAA is observed for the baseline within $0\lessapprox x/D\lessapprox 0.75$ and $\pm 0.5\lessapprox y/D\lessapprox \pm 1$, followed by velocity overshoot near the separated shear layers. The LB41, however, presents a broader FAA within $0\lessapprox x/D\lessapprox 1.5$ and $\pm 0.5\lessapprox y/D\lessapprox \pm 1.2$ with much more streamlined behaviour than that of the baseline, which is consistent with the delay in vortex shedding. Therefore, LB41 restrains the entrainment of high-momentum fluid from the free stream into the wake near the base of the cylinder, resulting in lower momentum transfer and slower vertical flow movement, as shown in figure 8( f). On the other hand, LB90 exhibits the opposite behaviour, where higher momentum transfer and faster vertical flow movement occur (see figure 8g). The size of the FAA observed for LB131 is smaller compared with LB41; being $0\lessapprox x/D\lessapprox 1.2$ and $\pm 0.5\lessapprox y/D\lessapprox \pm 1$, suggesting that vortex-shedding mechanism should be different from that of LB41. Indeed, the discharge of flow into the shear layers in the case of LB131 resembles a vortex-shedding mechanism commonly observed in porous materials.

To more accurately categorise the vortex-formation regions, the vortex core length, $L_{vc}$, the circulation length, $L_{r}$, and the vortex formation length, $L_{f}$, marked by the circle, square and triangle symbols in figure 8, respectively, are estimated from the distributions of $\bar {u}/U_{\infty }$ and the r.m.s. of the vertical velocity fluctuations, $v'_{{rms}}/U_{\infty }$, along $y/D=0$ as presented in figures 9(a) and 9(b), respectively. These locations for the baseline and LB cases are also provided in table 3. As demonstrated by Williamson (Reference Williamson1996), $L_{vc}$ corresponds to the core position of the mean recirculation region, while $L_{r}$ is defined as the streamwise length of the saddle point, where the flow field from outside the wake crosses the centreline. As shown in figure 9(a), the $x$-position of the maximum velocity deficit, $\bar {u}_d/U_{\infty }$, or minimum (dip) $\bar {u}/U_{\infty }$ can be an estimate of $L_{vc}$. Therefore, a strong reverse flow occurs in the centre of main recirculation region (vortex cores) as previously observed in figure 8. For the baseline, it is observed that $L_{vc}$ is around $0.75D$ when $\bar {u}_{d}/U_{\infty }\approx -0.1$. For LB41 and LB131, stronger reverse flow ($\bar {u}_{d}/U_{\infty }\approx -0.2$ and $-0.12$, respectively) occurs at $L_{vc}\approx 1.5D$, which indicates that the application of LB cases in the boundary layer and separated shear layers, respectively, moves the vortex cores farther downstream. In contrast, LB90 pulls the vortex cores slightly upstream by approximately $0.05D$ relative to the baseline case.

Figure 9. (a) Normalised time-averaged streamwise velocity, $\bar {u}/U_{\infty }$, and (b) the root-mean-square (r.m.s.) of the vertical velocity fluctuations, $v'_{{rms}}/U_{\infty }$, along the centreline ($\kern0.08em y/D=0$). The solid circle, square and triangle symbols represent $x\approx L_{vc}$, $L_{r}$ and $L_{f}$, respectively. The vertical dashed grey lines indicate the base of the cylinder at $x/D=0.5$.

Table 3. The vortex core length, $x\approx L_{vc}$, the length of the recirculation region, $x\approx L_{r}$, the length of the vortex-formation region, $x\approx L_{f}$, and maximum centreline velocity deficit, $\bar {u}_d/U_{\infty }$, for the baseline and LB cases. The results are extracted from figure 9.

The $\bar {u}/U_{\infty }$ profile can also be used to estimate $L_{r}$ as a streamwise location where $\bar {u}/U_{\infty }=0$ (He, Li & Wang Reference He, Li and Wang2014), corresponding to the end of the recirculation region. For the baseline, $L_{r}$ levels out at a value of approximately $1.2D$ in the upper end of the subcritical regime (Norberg Reference Norberg1987), which is consistent with the findings of the present experimental study, as shown in table 3. The $L_{r}$ values clearly show that the recirculation bubbles for each LB except LB90 are more extended along the streamwise direction in comparison with the baseline. The results show while both LB41 and LB131 present vortex cores at $L_{vc}\approx 1.5D$, LB41 has larger recirculation bubbles compared with LB131. The recirculation bubbles are observed within $0.5\lessapprox x/D \lessapprox 2.3$ and $0.7\lessapprox x/D \lessapprox 2.1$ in the cases of LB41 and LB131, respectively, as depicted in figure 8. The discrepancy in the recirculation patterns between LB131 and LB41 can potentially be attributed to a reduction in reverse flow observed in the case of LB131 (see figure 9a). This mechanism will be further discussed and elaborated on in the upcoming detailed analysis in § 5.4.

The length $L_{f}$ is defined as the distance between the cylinder centre and the end of the vortex formation region, where the maximum r.m.s. of the vertical velocity fluctuations are obtained along $y/D=0$ (see figure 9b). Bevilaqua (Reference Bevilaqua1973) demonstrated that an increase in $v'_{{rms}}/U_{\infty }$ is associated with the presence of large swirling turbulent structures, i.e. the formation of vortex shedding, and a highly anisotropic flow. As shown in figure 9(b), LB90 exhibits the maximum $v'_{{rms}}/U_{\infty }$ at $L_{f}\approx 1D$, which is approximately 0.5$D$ closer to the cylinder than the baseline case. As a result, LB90 generates a more energetic and turbulent flow field in the vicinity of the cylinder when compared with the baseline case. At $x/D\gtrapprox 1$, however, LB90 shows a reduction in $v'_{{rms}}/U_{\infty }$ from that of the baseline. As anticipated, other LB cases yield greater $L_{f}$ values relative to the baseline and are consistent with a reduction in $v'_{{rms}}/U_{\infty }$ magnitude close to the cylinder. The LB41 reveals that the vortex-formation region ends at $L_{f}\approx 2.6D$, which is the maximum length among all LB cases (see table 3). Furthermore, LB41 demonstrates a notable reduction in vertical velocity fluctuations across the wake region.

Figure 10 presents the profiles of $\bar {u}/U_{\infty }$ and $\bar {v}/U_{\infty }$ at different locations downstream of the cylinder, namely $x/D=0.7$, 1, 1.5, 2, 2.5 and 4.5. For a typical cylinder, the $\bar {u}/U_{\infty }$ profile varies gradually from a ‘top hat’ to a Gaussian-type as $x/D$ increases, which is due to the entrainment of the high-velocity fluid from the outer region into the wake (Heidari Reference Heidari2016). The fluid entrainment also increases the streamwise velocity magnitude in the inner region as one moves downstream behind the cylinder, particularly along the axis of symmetry. The wake region corresponding to the ‘top hat’-type profile initiates just behind the cylinder ($x/D=0.5$) with a width of approximately $1D$ ($-0.5\lessapprox y/D\lessapprox 0.5$) and ends at a downstream location where a transition to a Gaussian-type profile starts. In the ‘top hat’-type profile, the inner region is characterised by a minimum constant velocity, indicating a laminar flow with minimal turbulence. At $x/D=0.7$ as shown in figure 10(ai), the $\bar {u}/U_{\infty }$ profile of LB90 is a Gaussian-type yet other LB cases present a ‘top hat’-type profile. Gaussian-type $\bar {u}/U_{\infty }$ profiles for other LB cases are observed away from the base of the cylinder at $x/D=1.5$ (see figure 10aiii). The proximity of $x/D=0.7$ and $1.5$ to $L_{vc}$ for LB90 and other LB cases, respectively (as shown in table 3), suggests that (i) the wake region with a Gaussian-type profile originates from the vortex cores and (ii) LB90 exhibits a higher fluid entrainment compared with the other LB cases, as will be discussed in more detail in § 5.4.

Figure 10. Normalised time-averaged $x$- and $y$-components of velocity, (ai–vi) $\bar {u}/U_{\infty }$ and (bi–vi) $\bar {v}/U_{\infty }$, respectively. The selected stations correspond to $x\approx L_{vc}$, $L_{r}$ and $L_{f}$ of the baseline and LB cases as shown in figure 9.

From figure 10(bi–v), it is observed that the significant variation of $\bar {v}/U_{\infty }$ in the wake of each case within $-0.5\lessapprox y/D \lessapprox 0.5$ occurs between the values of $L_{vc}$ and $L_{r}$ or $L_{f}$ for each case. The observed variation can be ascribed to the intricate interaction between shear layers as part of the fluid entrainment process in the wake. This occurs when high-momentum fluid from the free stream penetrates the wake, leading to a region of diminished vertical velocity near the wake centreline. However, as the flow progresses away from the centreline towards the edges of the separated shear layers ($\kern0.08em y/D \approx \pm 0.5$), the velocity experiences an increase. It is observed that a very rapid vertical flow movement occurs within $0.7\leq x/D \leq 1$ for LB90 (see figure 10bi,ii) or within $1.5\leq x/D \leq 2.5$ for LB41 (see figure 10biii,v). While some LB cases exhibit similar $\bar {u}_{d}/U_{\infty }$ values (such as LB41 and LB90, as shown in table 3), the significant variation in $\bar {v}/U_{\infty }$ within the range of $L_{vc}\lessapprox x \lessapprox L_{r}$–$L_{f}$ is more pronounced for LB90. For LB90, the variation range of $\bar {v}/U_{\infty }$ is approximately $-0.2\leq \bar {v}/U_{\infty } \leq 0.2$, whereas for LB41 cases, it is around $-0.15\leq \bar {v}/U_{\infty } \leq 0.15$. In the case of LB131, the $\bar {v}/U_{\infty }$ exhibits variations approximately between $-0.12$ and 0.12, as depicted in figure 10(biii,iv).

Based on the analysis of near-field pressure, far-field noise and flow characteristics, it can be concluded that the vertical flow movement plays a significant role in the generation of vortex shedding noise. The observations in figure 8 indicate that all LB cases, except LB90, delay the vortex shedding and promote the farther-downstream roll-up of shear layers. This delay in vortex shedding is consistent with the observed base pressure recovery (Showkat Ali et al. Reference Showkat Ali, Mahdi and Da Silva2018) and reduced momentum transfer near the cylinder. Consequently, the vertical movement of the flow near the cylinder is reduced, leading to the suppression of surface pressure fluctuations, lift fluctuations (see figures 6 and 7, respectively) and a reduction in vortex shedding noise (see figure 5). These findings align with the stabilisation of the near wake and the development of a more streamlined flow pattern, as evident from the flow visualisation in figure 8.

Figure 11 shows the normalised wake width, $W/D$, defined as where the velocity is restored to half of $\bar {u}_d/U_{\infty }$ (He et al. Reference He, Li and Wang2014; Duan & Wang Reference Duan and Wang2021). The $W/D$ depends primarily on the scale of the Kármán vortices in the wake (Duan & Wang Reference Duan and Wang2021). As the periodic entrainment of the shear layers causes them to move towards the centreline, the wake width decreases immediately behind the cylinder to reach a minimum within $x\approx L_{vc}$–$L_{r}$ for each case. The wake width then vertically expands for $x\gtrapprox L_{vc}$–$L_{r}$ as a result of turbulent mixing (He et al. Reference He, Li and Wang2014). As reported by Duan & Wang (Reference Duan and Wang2021), the occurrence of intense turbulent mixing within the wake is attributed to the simultaneous acceleration of wake vortices to the convective velocity and their mutual interaction. It is observed that LB90 displays a narrower wake compared with the baseline up to $x/D\approx 1.5$. However, beyond this point, the wake width of LB90 increases, suggesting the generation of larger coherent flow structures compared with the baseline. On the contrary, for other LB cases, particularly LB41, the opposite trend is observed compared with LB90. Up to $x/D\approx 2.6$, where the vortex-formation region ends in the case of the LB41, LB41 exhibits a wider wake in comparison with LB131. This observation further supports the notion of distinct recirculation regions and different vortex-shedding mechanisms between LB41 and LB131.

Figure 11. Variation of the normalised wake width, $W/D$, with respect to $x/D$. Curves in this figure are fitted splines to the original data, and the vertical dashed grey line indicates the base of the cylinder at $x/D=0.5$. The wake centreline corresponds to $W/D=0$.

4.2. Shear layer trajectory

The propagation of the shear layer into the wake, as a result of LB, is investigated here. The trajectory of the shear layer is estimated by taking the averaged $y$-coordinate of the normalised time-averaged vorticity field ($\omega ^{*}=\bar {\omega }_{z}D/U_{\infty }\geq 0.75$), at each $x$-coordinate. The vorticity regions (identified as grey dots), and shear layer trajectories (as a coloured line that passes through these dots) are shown as subplots in figure 12 for each case.

Figure 12. Shear layer trajectory from the bottom side of the cylinders. Contour plots represent the normalised time-averaged vorticity, $\omega ^{*}=\bar {\omega }_{z}D/U_{\infty }$, and grey dots depict regions where $\omega ^{*}\geq 0.75$. The square symbols indicate the end of the shear layer at $x\approx L_{s}$ and $y\approx W_{s}$. The curves in this figure are fitted splines to the original data.

For the baseline case, it is observed that the trajectory line comes to an end at approximately $x/D \approx 1.2$. This observation confirms that the length of the turbulent shear layer, $L_{s}$, is nearly equal to the recirculation length, i.e. $L_{s} \approx L_{r}$. At $x\approx L_{s}$, the trajectory of the shear layer is closest to the centreline, denoted as $y\approx W_{s}$, which is consistent with the minimum width of the wake depicted in figure 11. The strong interaction between the upper and lower shear layers, therefore, would tend to occur at this location, where Gerrard (Reference Gerrard1966) reported that the shear layer thickness or diffusion length, affects the vortex-shedding frequency.

For LB90, both $L_{s}$ and $W_{s}$ decrease compared with the baseline, indicating a more intense interaction between the shear layers that occurs farther upstream. This observation is consistent with (i) the increased turbulence in the oscillating near wake just behind the cylinder (refer to § 4.3) and (ii) the higher broadband energy content of vertical velocity fluctuations (refer to § 4.4). In contrast, the other LB cases show larger $L_{s}$ and $W_{s}$ compared with the baseline, with LB41 demonstrating a more pronounced effect in this regard.

4.3. Production of energy and turbulence

Figure 13 presents the normalised $x$- and $y$-components of Reynolds stress, $\overline {u'u'}/U^{2}_{\infty }$ and $\overline {v'v'}/U^{2}_{\infty }$, respectively, and shear stress $\overline {u'v'}/U^{2}_{\infty }$. The $\overline {u'u'}/U^{2}_{\infty }$ and $\overline {v'v'}/U^{2}_{\infty }$ represent the variation in turbulence kinetic energy (TKE) across the wake, while $\overline {u'v'}/U^{2}_{\infty }$ depicts the flux of momentum transported by turbulent fluctuations and is also relevant to the acoustic analysis (Ffowcs Williams & Hall Reference Ffowcs Williams and Hall1970).

Figure 13. (a–d) Normalised time-averaged $x$-component of Reynolds stress, $\overline {u'u'}/U^{2}_{\infty }$, (e–h) normalised time-averaged $y$-component of Reynolds stress, $\overline {v'v'}/U^{2}_{\infty }$ and (i–l) normalised time-averaged shear stress, $\overline {u'v'}/U^{2}_{\infty }$. For each case, the solid circle, square and triangle symbols represent $x\approx L_{vc}$, $L_{r}$ and $L_{f}$, respectively, (see figure 9).

In the baseline case, the Reynolds stress $\overline {u'u'}/U^{2}_{\infty }$ exhibits dual peak regions. These peaks are observed at $y/D\approx \pm 0.5$ within the separated shear layers, and within the range of $0.75\lessapprox x/D \lessapprox 1.2$–$1.5$ in the high-momentum transfer area between $L_{vc}$ and $L_{r}$–$L_{f}$. The dual peaks serve as indicators of the concentration of vortical structures within the shear layers and the enhanced turbulent fluctuations and momentum transfer within the high-momentum transfer region. These regions experience complex flow dynamics, with interactions between the separated flow and surrounding fluid leading to increased turbulence, vorticity and mixing. The maximum value of $\overline {v'v'}/U^{2}_{\infty }$, however, is observed at $y/D=0$ and $x/D\approx 1.2$–$1.5$, suggesting that the momentum transfer beyond the vortex core centre ($x\approx 0.75$), which occurs in the vicinity of the centreline, contributes to an increase in the vertical periodic fluctuations. These fluctuations play a role in separating the vortices concentrated in the tails of the shear layers during the alternate shedding of the vortices at the end of the vortex-formation region. The interaction between the vortices and the surrounding flow in this region leads to enhanced turbulence and momentum exchange, resulting in the observed maximum $\overline {v'v'}/U^{2}_{\infty }$ on the centreline.

The results demonstrate that beyond $x\approx 0.75$, the magnitude of $\overline {v'v'}/U^{2}_{\infty }$ is higher than that of $\overline {u'u'}/U^{2}_{\infty }$, which can be attributed to two factors. Firstly, the streamwise velocity component aligns with the mean flow direction, resulting in a more organised and coherent flow structure. This alignment leads to relatively weaker turbulent fluctuations in the streamwise direction, causing a lower magnitude of $\overline {u'u'}/U^{2}_{\infty }$. Secondly, the shedding of vortices in the wake is primarily driven by the shear layers instabilities and the roll-up of the shear layers. These instabilities are more prominent in the vertical direction, leading to stronger vertical velocity fluctuations and a higher magnitude of $\overline {v'v'}/U^{2}_{\infty }$.

As shown in figure 13, the LB cases exhibit maximum values of $\overline {u'u'}/U^{2}_{\infty }$ and $\overline {v'v'}/U^{2}_{\infty }$ within the ranges of $L_{vc}\lessapprox x \lessapprox L_{r}$–$L_{f}$ and $x\approx L_{r}$–$L_{f}$, respectively, similar to the baseline. From figure 14(ai–vi), it can be observed that each LB case leads to a reduction in $\overline {u'u'}/U^{2}_{\infty }$, particularly in the dual peak regions up to their respective $L_{vc}$. The LB41 case exhibits a more significant reduction compared with the other LB cases. However, no significant decrease is observed for LB90, which could be attributed to the smaller size of the vortex-formation region in this case.

Figure 14. (ai–vi) Normalised time-averaged $x$-component of Reynolds stress, $\overline {u'u'}/U^{2}_{\infty }$, (bi–vi) normalised time-averaged $y$-component of Reynolds stress, $\overline {v'v'}/U^{2}_{\infty }$ and (ci–vi) normalised time-averaged shear stress, $\overline {u'v'}/U^{2}_{\infty }$, at different $x$-locations corresponding to $x\approx L_{vc}$, $L_{r}$ and $L_{f}$ for the baseline and LB cases as shown in figure 9.

The quantity $\overline {v'v'}/U^{2}_{\infty }$ is also decreased by each LB, particularly along $y/D=0$. However, the trends with respect to $x/D$ vary among the different LB cases, as shown in figure 14(bi–vi). This indicates that each LB affects the vortex shedding process through different mechanisms. The application of LB41 within the boundary layer causes the shear layers to stretch and roll up farther downstream, which results in a reduction in $\overline {v'v'}/U^{2}_{\infty }$ across the entire range of $x/D$, as observed in figure 9(b). In this case, the most significant performance is observed near the base of the cylinder, and the level of reduction decreases as $x/D$ increases. In the case of LB90, where the blowing is applied perpendicular to the free stream at the cylinder shoulders just beyond the separation point, a substantial increase in $\overline {v'v'}/U^{2}_{\infty }$ is observed near the base of the cylinder, extending up to $x/D=1$ ($x\approx L_{r}$ or $L_{f}$). This increase is attributed to the rapid vertical flow movement during rapid rolling up of the shear layer, which occurs farther upstream in this case (see figures 9b and 10bi–vi). While it is not significant, LB90 suppresses $\overline {v'v'}/U^{2}_{\infty }$ for $x/D>1$. In the case of LB131, the separated shear layers are directly influenced by the blowing vortices, and their behaviour regarding $\overline {v'v'}/U^{2}_{\infty }$ is opposite to LB90. It should be noted that the transition point ($x\approx L_{r}$ or $L_{f}$) moves downstream for LB131, occurring at $x/D>2$. From figure 14(avi,bvi), it is evident that at $x/D=4.5$ and $y/D=0$, $\overline {u'u'}/U^{2}_{\infty }$ remains lower than $\overline {v'v'}/U^{2}_{\infty }$. This observation suggests that the flow in this region still exhibits anisotropic characteristics and has not fully transitioned to the far wake with a more isotropic state. However, LB41 shows a greater tendency to approach this isotropic state compared with the other cases. This is evident from the fact that $\overline {v'v'}/U^{2}_{\infty }$ for LB41 is approximately one and a half times as high as $\overline {u'u'}/U^{2}_{\infty }$, while in the other cases it is two and a half times higher. This suggests that LB41 exhibits a stronger development towards isotropy, with a more balanced distribution of turbulence intensity between the streamwise and vertical directions.

From figure 13(i–l), it can be observed that $\overline {u'v'}/U^{2}_{\infty }$ exhibits a significant variation within the range of $-0.5\lessapprox y/D\lessapprox 0.5$, while $x$ varies approximately from $L_{vc}$ to $L_{r}$–$L_{f}$. This behaviour is similar to the observed variation of $\bar {v}/U_{\infty }$ in figure 8(e–h). The development and interaction of the shear layers with the surrounding flow result in intensified velocity gradients and vorticity. This intensification leads to enhanced turbulence and vorticity, causing increased flow fluctuations and the generation of shear stress. The concentration and interaction of vortices within the shear layers further amplify the turbulence and momentum transfer. These vortices create coherent structures and eddies that shed at the end of the vortex-formation region, thereby contributing to higher shear stress levels.

It is observed in figure 14(ci–vi) that the $x/D$-range where $\overline {u'v'}/U^{2}_{\infty }$ maximises in the cases of LB41 and LB90 is $1.5\leq x/D \leq 2.5$ and $0.7\leq x/D \leq 1$, respectively. These observations align with the patterns seen in figure 10(bi–vi) and indicate that the rapid vertical movements of the flow are driven by the momentum carried by the turbulent fluctuations. At a distance far away from the cylinder, specifically at $x/D=4.5$, the profiles of $\overline {u'v'}/U^{2}_{\infty }$ appear relatively flat. This indicates that the flow fields in the wake tend to be nearly uniform, with little variation in the momentum flux. In this region, the turbulent fluctuations have dissipated, resulting in a more stable and less dynamic flow. By inspecting figure 14(ci–vi), it is clear that all LB cases, except for LB90, result in a reduction of $\overline {u'v'}/U^{2}_{\infty }$ compared with the baseline, but this reduction is only observed up to $x/D=2$. However, it should not be assumed that an increase in $\overline {u'v'}/U^{2}_{\infty }$ by LB41 and LB131 at $x/D\geq 2$ would lead to an increase in tonal noise relative to the baseline. The proximity of the vortex-formation region to the cylinder plays a role in influencing the noise reduction achieved by LB techniques. In the case of LB90, vortex shedding occurs rapidly, resulting in a significant vertical variation of $\overline {u'v'}/U^{2}_{\infty }$ close to the cylinder. This variation promotes lift fluctuations and leads to increased far-field noise. On the other hand, other LB cases, particularly LB41, cause a delay in vortex shedding, resulting in a significant variation of $\overline {u'v'}/U^{2}_{\infty }$ farther downstream. This is consistent with the emergence of a weaker hydrodynamic field with lower energy content near the cylinder. The reduction of flow-induced noise by LB techniques is also influenced by factors such as TKE and the scale of the most energetic fluctuations in the vicinity of the cylinder. These factors play a role in shaping the hydrodynamic field and the associated noise characteristics.

Figure 15 shows that the maximum $\mathrm {TKE^{*}}=(\overline {u'u'}+\overline {v'v'})/(2U_{\infty }^{2}$) appears at $y/D=0$ and $x\approx L_{r}$–$L_{f}$ for each case, which is consistent with the observations in figure 14(bi–vi). Since the maximum $\overline {v'v'}/U^{2}_{\infty }$ (see figure 14bi–vi) is significantly greater than that of $\overline {u'u'}/U^{2}_{\infty }$ (see figure 14ai–vi), $\mathrm {TKE^{*}}$ is dominated by vertical fluctuations as demonstrated by others (Feng & Wang Reference Feng and Wang2010; He et al. Reference He, Li and Wang2014; Octavianty & Asai Reference Octavianty and Asai2016; Qu et al. Reference Qu, Wang, Sun, Feng, Pan, Gao and He2017; Chauhan et al. Reference Chauhan, Dutta, More and Gandhi2018; Sharma & Dutta Reference Sharma and Dutta2020). It is observed for the baseline that the intense $\mathrm {TKE^{*}}$ region ($\mathrm {TKE^{*}}\gtrapprox 0.3$) is larger in vertical scale compared with that of $\overline {v'v'}/U^{2}_{\infty }$. This discrepancy can be related to the complex interactions between the shear layers and the entrained fluid, which generate significant shear and vorticity, promoting turbulence production. In addition, the energy cascade process allows for the generation of vortices and turbulent eddies, which contribute to the overall vertical scale of turbulence in the wake.

Figure 15. Normalised TKE, $\mathrm {TKE^{*}}$, for the baseline and LB cases. For each case, the solid circle, square and triangle symbols represent $x\approx L_{vc}$, $L_{r}$ and $L_{f}$, respectively (see figure 9).

While the $\mathrm {TKE^{*}}$ of LB90 is lower in magnitude compared with the baseline, which is consistent with the observation in figure 13(e,g), the rapid rolling up of the shear layers concentrates the vorticity near the cylinder (smaller vortex formation size), which facilitates their mutual interaction to increase $\mathrm {TKE^{*}}$ in this region. For other LB cases a delay in vortex shedding elongates the shear layers and impedes their interaction, thus $\mathrm {TKE^{*}}$ decreases close to the cylinder. Overall, the vortex formation length and the interaction between the shear layers determine the level of $\mathrm {TKE^{*}}$, which is a measure of turbulence mixing (Benard et al. Reference Benard, Balcon, Touchard and Moreau2008; Schäfer, Breuer & Durst Reference Schäfer, Breuer and Durst2009; Feng & Wang Reference Feng and Wang2010).

4.4. Velocity fluctuations power spectra in the wake

Figures 16 and 17 present the PSD of the $x$- and $y$-components of velocity fluctuations, $\varPhi _{uu}$ and $\varPhi _{vv}$ (dB Hz$^{-1}$, ref: $1\,{\rm m}\,{\rm s}^{-1}$), respectively. Upon observing the figures, a distinct tonal peak at the frequency of $f_{1}$ is noticeable for each case. This peak aligns closely with the $f_{1}$-tone characteristics depicted in figures 5 and 6, which suggests that the pronounced far-field noise and near-field pressure, respectively, can be attributed to the periodic shedding of vortices. In the case of the baseline, the $f_{1}$-tone achieves its maximum magnitude at $x/D=1$ and 1.5 in the $\varPhi _{uu}$ and $\varPhi _{vv}$ spectra, as illustrated in figures 16( f) and 17( f), respectively. It is noteworthy that in this scenario, the shear layers undergo roll-up at $L_{r}\approx 1$ and subsequently shed vortices at $L_{f}\approx 1.5$ (see figure 9).

Figure 16. Streamwise velocity PSD, $\varPhi _{uu}=10\log _{10}\,{\phi _{uu}}$. (a–e) Isosurface of $\varPhi _{uu}$ where transparency decreases with increasing $\varPhi _{uu}$ and ( f–i) the $\varPhi _{uu}$ values along $a$ ($\kern0.08em y/D=0$) and $b$ ($\kern0.08em y/D=0.5$) for stations A ($x/D=0.7$), B ($x/D=1$), C ($x/D=1.5$) and E ($x/D=2.5$).

Figure 17. Vertical velocity PSD, $\varPhi _{vv}=10\log _{10}\,{\phi _{vv}}$. (a–e) Isosurface of $\varPhi _{vv}$ where transparency decreases with increasing $\varPhi _{vv}$ and ( f–i) the $\varPhi _{vv}$ values along $a$ ($\kern0.08em y/D=0$) and $b$ ($\kern0.08em y/D=0.5$) for stations A ($x/D=0.7$), B ($x/D=1$), C ($x/D=1.5$), D ($x/D=2$) and E ($x/D=2.5$).

It is observed from figure 16(a,b, f) that the $\varPhi _{uu}$ spectra of the baseline are purely broadband at the $f_{1}$-tone frequency along the centreline. Away from $y/D=0$, however, they include the $f_{1}$-tone, which increases in magnitude with $y/D$ to reach a maximum at $y/D\approx 0.5$, where in the separated shear layers. In contrast, the $\varPhi _{vv}$ spectra present the $f_{1}$-tone at all measurement points across the wake yet decrease in magnitude within $y/D\approx 0$–$0.5$ as shown in figure 17(a,b, f). These findings indicate that the streamwise and vertical velocity fluctuations in the separated shear layers and along the centreline, respectively, contribute significantly to the vortex shedding, supporting the results of $\overline {u'u'}/U^{2}_{\infty }$ and $\overline {v'v'}/U^{2}_{\infty }$ (see figures 13 and 14) and their role in TKE production (see figure 15). However, the vertical velocity fluctuations exhibit a stronger influence on vortex shedding, as indicated by the larger magnitudes of the $f_{1}$-tone in the $\varPhi _{vv}$ spectra compared with the $\varPhi _{uu}$ spectra.

In addition to the $f_{1}$-tone, the first two harmonics are observed in figures 16( f) and 17( f), which indicate the footprint of velocity fluctuations in lift and drag oscillations of the cylinder. The $\varPhi _{uu}$ spectra along $y/D=0$ show non-negligible tonal peaks at the $f_{2}$-tone frequency in agreement with Showkat Ali et al. (Reference Showkat Ali, Mahdi and Da Silva2018), whereas these tonal peaks appear in the $\varPhi _{vv}$ spectra along $y/D=0.5$ beyond $x/D=1.5$. It is observed that the $f_{3}$-tone tends to appear only in the $\varPhi _{vv}$ spectra at $x/D=1.5$. As mentioned in § 3.3, lift and drag oscillate in odd and even harmonics, respectively. In this context, vertical velocity fluctuations, particularly at $x\approx L_{f}$ and $y/D=0$, play a significant role in generating lift fluctuations. This further supports the previous argument that high-momentum transfer, which affects the vertical flow movement, contributes to the fluctuations in lift.

From figure 16( f–i), it can be observed that each LB case, particularly LB41, is effective in reducing the magnitude of the $f_{1}$-tone in the $\varPhi _{uu}$ spectra up to $x/D=1.5$. Similarly, all LB cases lead to a decrease in the magnitude of the $f_{1}$-tone in the $\varPhi _{vv}$ spectra across the range of $x/D$, as shown in figure 17( f–i). However, LB90 shows an increase in the magnitude of the $f_{1}$-tone in the $\varPhi _{vv}$ spectra compared with the baseline, reaching up to $x/D=1$. The observations of $\overline {v'v'}/U^{2}_{\infty }$ in figure 14(bi–vi) align with these findings and support the conclusion that reducing streamwise velocity fluctuations through LB does not necessarily lead to the suppression of dominant vortex structures.

4.5. Near-field pressure to far-field noise coherence

The emergence of a tonal peak at the $f_{1}$-tone frequency in both spectra of near-field pressure and far-field noise demonstrated the role of surface pressure fluctuations in the tonal noise generation mechanism. To gain a better understanding of the far-field propagation effects of the identified surface pressure fluctuations, the coherence between the near-field pressure and the far-field noise was calculated as

(4.1)\begin{equation} \gamma^{2}_{pp_{_s}}=\frac{{\lvert \phi_{pp_{_s}}(f) \rvert }^{2}}{\phi_{pp}(f)\phi_{p_{_s}p_{_s}}(f)}, \end{equation}

where $\phi _{pp_{_s}}$ denotes the cross-spectrum between the acoustic and pressure signals, i.e. $p'$ and $p'_{s}$, respectively, and $\phi _{pp}$ and $\phi _{p_{_s}p_{_s}}$ are the autospectrum of each individual signal.

Figure 18 reveals that the baseline possesses high coherence at the $f_{1}$-tone frequency with broadband levels below 0.1. This finding provides support for the mechanism that tonal noise is primarily generated by the near-field pressure fluctuations caused by vortex shedding (Maryami et al. Reference Maryami, Arcondoulis, Liu and Liu2023c). In the baseline case, the $f_1$-tone magnitude is observed to be approximately 0.3 at $\theta =0^\circ$ and increases rapidly to around 0.65 at $\theta =41^\circ$. The magnitude then remains relatively constant within the range of $\theta =41^\circ$ to $\theta =131^\circ$ before dropping to zero at $\theta =180^\circ$. This trend highlights the significant contribution of surface pressure fluctuations, induced by vortex shedding in the preseparation and postseparation regions, to the generation of tonal noise (Maryami et al. Reference Maryami, Arcondoulis, Liu and Liu2023c). Previous studies have reported that surface pressure fluctuations near the flow separation point act as sound sources (Oguma et al. Reference Oguma, Yamagata and Fujisawa2013) and strongly scatter sound in a region approximately from $\theta =25^\circ$ to $155^\circ$ (Li, Rival & Wu Reference Li, Rival and Wu2021).

Figure 18. Coherence between the near-field pressure and far-field noise, $\gamma ^{2}_{pp_{_s}}$, in terms of $St$ for the pressure taps located at $\theta =0^{\circ }$, $41^{\circ }$, $90^{\circ }$, $131^{\circ }$ and $180^{\circ }$. The labels $f_{1}$ and $f_{2}$ represent the fundamental vortex shedding tone (first harmonic) and second harmonic, respectively.

All LB cases exhibit approximately a trend analogous to that of the baseline concerning the variation of the $f_{1}$-tone magnitude in relation to peripheral angles. Within the preseparation and postseparation regions (i.e. $\theta =41^\circ$–$131^\circ$), only LB41 exhibits a decrease in the magnitude of the $f_{1}$-tone compared with the baseline. This observation indicates that vortex shedding is effectively suppressed by LB41, leading to a dissipation of its energy content over a larger vortex-formation region (Suzuki & Colonius Reference Suzuki and Colonius2006). As a result, the vortex shedding does not generate a strong feedback signal to the cylinder, thereby influencing the surface pressure fluctuations and propagating strong acoustic waves to the far field. Additionally, LB41 shows a dual tonal peak around the $f_{1}$-tone frequency, which is consistent with the far-field and near-field pressures observed in figures 5 and 6, respectively.

4.6. Velocity-pressure coherence

To investigate the impact of velocity fluctuations on the scattering of acoustic waves into the far field, the coherence relationship between the velocity fluctuation and sound pressure signals is computed using

(4.2)\begin{equation} \gamma^{2}_{up}=\frac{{\lvert \phi_{up}(f) \rvert }^{2}}{\phi_{uu}(f)\phi_{pp}(f)}, \end{equation}

where $\phi _{up}(f)$ denotes the cross-PSD function between the velocity and sound pressure signals.

Figures 19 and 20 illustrate the coherence between the $x$- and $y$-components of velocity and far-field noise, represented by $\gamma ^{2}_{up}$ and $\gamma ^{2}_{vp}$, respectively. As expected, $\gamma ^{2}_{up}$ and $\gamma ^{2}_{vp}$ for the baseline present a strong tonal peak at the $f_{1}$-tone frequency, regardless of the measurement point in the wake. This confirms that velocity fluctuations responsible for vortex shedding are the primary mechanism of tonal noise. At the $f_{1}$-tone frequency along the centreline of the baseline, $\gamma ^{2}_{up}$ remains below 0.1. However, within the separated shear layers, up to $x\approx L_{r}$–$L_{f}$, $\gamma ^{2}_{up}$ reaches a maximum value of approximately 0.6–0.65, as indicated in figure 19(b, f). Additionally, the maximum value of $\gamma ^{2}_{up}$ is observed at $y/D\gtrapprox 0.5$. The maximum $\gamma ^{2}_{up}$ follows the shear layer trajectory and ends at $x/D\approx 1.5$, indicating that the vortices within the shear layer are fed by the streamwise velocity fluctuations to shed into a vortex and scatter noise. In contrast, $\gamma ^{2}_{vp}$ exhibits a consistent presence at the $f_{1}$-tone frequency with magnitudes ranging from approximately 0.63 to 0.7 at all measurement points, particularly along the centreline as shown in figure 20(b, f). It is noted that the maximum values of $\gamma ^{2}_{vp}$ are slightly higher than those of $\gamma ^{2}_{up}$. Although this difference is not substantial, it implies that vertical velocity fluctuations are more likely to scatter acoustic waves to the far field from any position within the wake region. These observations support previous explanations regarding the influence of rapid vertical flow movement on tonal noise generation.

Figure 19. Streamwise velocity to far-field noise coherence ($\gamma ^{2}_{up}$). (a) Isosurface of $\gamma ^{2}_{up}$ where transparency decreases with increasing $\gamma ^{2}_{up}$. (b–e) The values of $\gamma ^{2}_{up}$ recorded in the $xy$-plane and ( f–i) the values of $\gamma ^{2}_{up}$ along $a$ ($\kern0.08em y/D=0$) and $b$ ($\kern0.08em y/D=0.5$) for stations A ($x/D=0.7$), B ($x/D=1$), C ($x/D=1.5$) and E ($x/D=2.5$).

Figure 20. Vertical velocity to far-field noise coherence ($\gamma ^{2}_{vp}$). (a) Isosurface of $\gamma ^{2}_{vp}$ where transparency decreases with increasing $\gamma ^{2}_{vp}$. (b–e) The values of $\gamma ^{2}_{vp}$ recorded in the $xy$-plane and ( f–i) the values of $\gamma ^{2}_{vp}$ along $a$ ($\kern0.08em y/D=0$) and $b$ ($\kern0.08em y/D=0.5$) for stations A ($x/D=0.7$), B ($x/D=1$), C ($x/D=1.5$) and E ($x/D=2.5$).

This interpretation holds true for all LB cases, with the exception of certain measurement points in the wake where some deviations from the observed trend may occur. From figures 19 and 20, it is evident that LB41 exhibits a greater potential than other LB cases to reduce the magnitude of the $f_{1}$-tone in the $\gamma ^{2}_{up}$ and $\gamma ^{2}_{vp}$ spectra compared with the baseline. While there is some reduction in the $f_{1}$-tone magnitudes from that of the baseline in the case of LB131 at certain measurement points, particularly outside of the separated shear layers, it is not as significant as in LB41. On the other hand, LB90 shows a similar magnitude of the $f_{1}$-tone compared with the baseline, and in some measurement points, it even surpasses the baseline's magnitude. The delayed vortex shedding caused by LB cases, particularly LB41, and the subsequent depletion of energy in the coherent flow structures can be considered as mechanisms contributing to the reduction in coherence level between velocity fluctuations and acoustic pressure fluctuations in the far field. A delay in vortex shedding depletes the momentum transfer in the region of vortex formation, resulting in a reduction of the energy content of the coherent flow structures. As a consequence, the hydrodynamic pressure field generated by velocity fluctuations may not have enough energy to produce strong acoustic waves that can propagate to the far field.

Figures 21 and 22 show the coherence between the $x$- and $y$-components of velocity and the near-field pressure measured around the cylinder circumference, $\gamma ^{2}_{up_{_s}}$ and $\gamma ^{2}_{vp_{_s}}$, respectively. For brevity, the results are only presented for surface pressure fluctuations at angular locations $\theta =0^{\circ }$, $41^{\circ }$, $90^{\circ }$, $131^{\circ }$ and $180^{\circ }$.

Figure 21. Isosurface of coherence between the $x$-component of velocity and near-field pressure, $\gamma ^{2}_{up_{_s}}$, where transparency decreases with increasing $\gamma ^{2}_{up_{_s}}$: (a) baseline; (b) LB41; (c) LB90; (d) LB131.

Figure 22. Isosurface of coherence between the $y$-component of velocity and near-field pressure, $\gamma ^{2}_{vp_{_s}}$, where transparency decreases with increasing $\gamma ^{2}_{vp_{_s}}$: (a) baseline; (b) LB41; (c) LB90; (d) LB131.

The results from each case demonstrate that both $\gamma ^{2}_{up_{_s}}$ and $\gamma ^{2}_{vp_{_s}}$ exhibit high values at the $f_{1}$-tone frequency. This behaviour is observed in various sections of the cylinder circumference, particularly in the region within $\theta =41^{\circ }$–$131^{\circ }$, suggesting that during intermittent vortex shedding, the cylinder shoulders experience significant pressure fluctuations. It was already shown in figure 18 that $\gamma ^{2}_{pp_{_s}}$ is maximised at the $f_{1}$-tone frequency in the preseparation and postseparation regions. Therefore, the results in figures 21 and 22 confirm that the pressure fluctuations induced by vortex shedding at the shoulders of the cylinder generate strong acoustic waves that propagate to the far field. This aligns with the viewpoint of Inoue & Hatakeyama (Reference Inoue and Hatakeyama2002) who demonstrated that pressure waves are produced by the alternating pressure pulses generated on both sides of the cylinder during vortex shedding.

The spatial distributions of $\gamma ^{2}_{up_{_s}}$ and $\gamma ^{2}_{vp_{_s}}$ at the $f_{1}$-tone frequency in the wake of each cylinder resemble those of $\gamma ^{2}_{up}$ and $\gamma ^{2}_{vp}$, respectively. This correspondence further supports the notion that surface pressure fluctuations and far-field noise are generated through the same underlying mechanism. Nevertheless, the magnitudes of the tonal peaks in $\gamma ^{2}_{up_{_s}}$ and $\gamma ^{2}_{vp_{_s}}$ are higher compared with those in $\gamma ^{2}_{up}$ and $\gamma ^{2}_{vp}$. This discrepancy can be attributed to the difference in the measurement point spacing. In the case of $\gamma ^{2}_{up_{_s}}$ and $\gamma ^{2}_{vp_{_s}}$, the maximum distance between measurement points is approximately $4.5D$, whereas it reaches approximately $23D$ for $\gamma ^{2}_{up}$ and $\gamma ^{2}_{vp}$. As a result, the localised measurements in the vicinity of the cylinder yield higher tonal peak levels. Figures 21 and 22 illustrate that $\gamma ^{2}_{vp_{_s}}\gtrapprox \gamma ^{2}_{up_{_s}}$ at the $f_{1}$-tone frequency for most measurement points, which is consistent with the observations of $\gamma ^{2}_{vp}\gtrapprox \gamma ^{2}_{up}$ in figures 19 and 20. This finding further supports the notion that vertical velocity fluctuations in the wake play a prominent role in influencing both the near-field pressure and far- field noise.

The effect of LB cases in suppressing vortex shedding manifests itself at the level of $\gamma ^{2}_{up_{_s}}$ and $\gamma ^{2}_{vp_{_s}}$ in a behaviour similar to that observed for $\gamma ^{2}_{up}$ and $\gamma ^{2}_{vp}$, respectively. However, at the front and rear stagnation points, especially $\theta =0^{\circ }$, the $\gamma ^{2}_{up_{_s}}$ and $\gamma ^{2}_{vp_{_s}}$ of LB cases are slightly higher than those of the baseline. It is unclear, yet it can be ascribed to the effect of feedback signal on the disturbance in the shear layers on the cylinder surface which modulates the boundary layer and affects pressure fluctuations at the stagnation point to correlate with the hydrodynamic field in the wake region.

The measurements of $\gamma ^{2}_{pp_{s}}$, $\gamma ^{2}_{up_{s}}$, $\gamma ^{2}_{vp_{s}}$, $\gamma ^{2}_{up}$ and $\gamma ^{2}_{vp}$ provide valuable insights into the mechanisms involved in tonal noise generation. It can be inferred that vortex shedding contributes to tonal noise through two distinct mechanisms: surface pressure fluctuations at the cylinder shoulders and the hydrodynamic field in the wake. Both mechanisms generate sound waves that can propagate to the far field, albeit with varying intensities of scattering. These findings highlight the complex nature of tonal noise generation and the role played by different components of the flow field. Comparing the results of $\gamma ^{2}_{pp_{_s}}$ and $\gamma ^{2}_{vp}$ at the $f_{1}$-tone frequency (see figures 18 and 20), it is observed that $\gamma ^{2}_{pp_{_s}}$ is approximately equal to $\gamma ^{2}_{vp}$ for the baseline case. However, for the LB cases and to some extent LB41, there is a difference where $\gamma ^{2}_{pp_{_s}}$ is greater than $\gamma ^{2}_{vp}$. This indicates that the acoustic waves resulting from surface pressure fluctuations are more likely to radiate to the far field compared with those originating from the hydrodynamic field. The findings are consistent with the study conducted by Li et al. (Reference Li, Rival and Wu2021), which emphasised that the shoulder of the cylinder plays a more significant role in sound emission than the oscillating near wake just behind the cylinder.

5.1. Proper orthogonal decomposition modes

Proper orthogonal decomposition (POD) is employed to analyse the coherent structures in the flow and determine the most energetic fluctuations, as well as their contributions to the dominant tonal peaks in the acoustic and velocity spectra. The orthogonal basis functions constructed using POD are optimal to identify dominant coherent structures and to construct reduced-order flow models that capture the largest amount of energy with the smallest number of modes (Sirovich Reference Sirovich1987). The POD analysis was conducted using all 5289 snapshots, following the procedure outlined by Chen et al. (Reference Chen, Reuss, Hung and Sick2013). Figure 23(a,b) present the energy fraction, $\lambda _{n}/\varSigma \lambda _{n}$, and cumulative energy, $\varSigma ^{n}_{1}\lambda _{n}/\varSigma \lambda _{n}$, respectively, for the first 100 modes, where $\lambda _{n}$ represents the eigenvalue associated with mode $n$. In all cases, the contribution of these modes to the total TKE ranges from 40 % to 60 %, as depicted in figure 23(b). In line with previous POD analyses conducted on wake flows with synthetic jet control (Feng, Wang & Pan Reference Feng, Wang and Pan2011) and jet flows (Kirby, Boris & Sirovich Reference Kirby, Boris and Sirovich1990), the initial modes of the POD analysis capture a significant amount of energy for all cases. However, the energy contribution decreases gradually for the higher modes, ultimately approaching zero, as depicted in figure 23(a). Previous studies conducted by Dipankar, Sengupta & Talla (Reference Dipankar, Sengupta and Talla2007) and Perrin et al. (Reference Perrin, Braza, Cid, Cazin, Barthet, Sevrain, Mockett and Thiele2007) have shown that the dominant features in the flow field are captured by the first few modes of the POD analysis, representing large-scale coherent structures. In contrast, the higher modes of the POD analysis are associated with smaller-scale turbulent structures.

Figure 23. (a) Energy fraction, $\lambda _{n}/\varSigma \lambda _{n}$, and (b) cumulative energy, $\varSigma ^{n}_{1}\lambda _{n}/\varSigma \lambda _{n}$, of the POD modes as a function of mode number, $n$, for the baseline and LB cases.

The measurement data contain temporal information that can be retrieved by the time coefficient, $c_{n}$, and projecting each snapshot onto the basis functions. The time coefficient indicates the correlation between the $n$th basis POD spatial mode and the instantaneous flow field. To delve deeper into the analysis of the dominant coherent structures, the energy spectra of the time coefficients, denoted as $\varPhi _{cc}$ (dB Hz$^{-1}$, ref : $1\,{\rm m}\,{\rm s}^{-1}$), for the first 100 POD modes are shown in figure 24. It is observed that the spectral shapes of the first two modes exhibit remarkable similarities across all cases. In addition, a prominent tonal peak is evident at the frequency corresponding to the $f_{1}$-tone, which confirms that the dominant coherent structures generated during vortex shedding, specifically the large-scale Kármán vortices, capture the majority of the energy (Dipankar et al. Reference Dipankar, Sengupta and Talla2007; Epps & Techet Reference Epps and Techet2010). Based on the characteristics of the far-field noise spectra depicted in figure 5, it can be inferred that the dominant coherent structures represented by modes 1 and 2 play a significant role in the generation of far-field noise. As the mode number increases, the energy content of the turbulent structures diminishes, indicating the presence of smaller-scale turbulent structures. These structures generally exhibit a broadband behaviour in the time coefficient spectra (see figure 24e–h) yet for certain modes, weak but discernible tonal peaks emerge at the $f_{2}$- and $f_{3}$-tone frequencies.

Figure 24. The PSD of the mode time coefficient, $\varPhi _{cc}=10\log _{10}\,{\phi _{cc}}$, as a function of $St$ and mode number ($n$). The labels $f_{1}$, $f_{2}$ and $f_{3}$ represent the first harmonic (fundamental vortex shedding tone), second and third harmonics, respectively.

From figure 24(e–h), it is evident that LB41 reduces the magnitude of the $f_{1}$-tone by approximately 6 dB compared with the baseline, while LB90 and LB131 exhibit an $f_{1}$-tone magnitude similar to that of the baseline ($\varPhi _{cc}\approx 56$ dB). These observations seem to contradict the findings in figure 5 and indicate that the energy content of the dominant coherent structures alone is not sufficient to evaluate their contribution to far-field noise. As discussed in §§ 4.5 and 4.6, the radiating acoustic waves resulting from surface pressure fluctuations strongly outweigh those from the hydrodynamic field in the cases of LB90 and LB131. This emphasises that the role of dominant coherent structures in far-field noise generation should be assessed based on their ability to induce surface pressure fluctuations, which is influenced by the size of the vortex-formation region as discussed in § 3.2. It is important to note that the hydrodynamic field experiences rapid attenuation with distance, as demonstrated by Suzuki & Colonius (Reference Suzuki and Colonius2006). This results in a weak feedback signal that has limited influence on the boundary layer modulation and disturbances in the shear layer, leading to a reduction in the energy content of surface pressure fluctuations, particularly at the $f_{1}$-tone frequency (refer to § 3.2). Therefore, regardless of the energy content of the dominant coherent structures, in the case of LB90, the reconstructed coherent structures of modes 1 and 2 appear near the cylinder and generate strong surface pressure fluctuations. However, for LB131, these coherent structures shed farther downstream, resulting in minimal impact on surface pressure fluctuations.

The high-frequency broadband noise observed in figure 5 may be influenced by the small-scale high-frequency fluctuations of mode 3. The energy source for these fluctuations is the large-scale coherent structures (Portela, Papadakis & Vassilicos Reference Portela, Papadakis and Vassilicos2020), which convect in the turbulent wake and dissipate energy. While the energy contents of the coherent and small-scale structures are of comparable magnitudes, the dissipation mechanisms for each type of motion are significantly different. Specifically, small-scale structures contribute more to energy dissipation compared with the coherent structures (Hussain Reference Hussain1983; Portela, Papadakis & Vassilicos Reference Portela, Papadakis and Vassilicos2018). In the case of LB131, where LB is applied within the separated shear layers, the small-scale structures of mode 3 convect rapidly and cause more intense energy dissipation within $St\approx 0.3$–$2$, which can be a mechanism for reducing high-frequency broadband noise (see figure 5). On the other hand, LB41 and LB90 are applied outside of the separated shear layers and do not stimulate the turbulent structures of mode 3 to dissipate energy effectively.

To gain a deeper understanding of the influence of coherent flow structures on near-field and far-field pressures, detailed autocorrelation and a wavelet transform are presented in figure 25. The autocorrelation determines the dominant time scales and is calculated as

(5.1)\begin{equation} R_{pp}(\tau)=\frac{\overline{p'(t+\tau)p'(t)}}{p^{2}_{_{{rms}}}}, \end{equation}

where $\tau$ is the time delay and the time average is represented by the overbar.

Figure 25. Autocorrelation of near-field pressure at $\theta =90^{\circ }$, $R_{p_{_s}p_{_{s}}}$, and far-field pressure, $R_{pp}$, with respect to normalised time delay, $\tau ^{*}=\tau U_{\infty }/D$, depicted by lines with and without symbols, respectively. The subplots present the real part of the wavelet coefficient, ${\rm I\!R}(W)$, applied to the time coefficient of mode 1 ($c_{1}$).

To investigate the amplitude modulation mechanism related to the coherent flow structures generated by vortex shedding, a wavelet transform was applied to the time coefficient of mode 1, denoted as $c_1$. This analysis allows us to analyse the modulation of amplitudes in both the time and frequency domains simultaneously. The continuous wavelet transform, denoted as $W(a,\tau )$, was utilised in this study, with the Morlet wavelet serving as the mother wavelet (Farge Reference Farge1992; Li & Nozaki Reference Li and Nozaki1995; Mallat Reference Mallat1999; Maryami et al. Reference Maryami, Ali, Azarpeyvand, Dehghan and Afshari2022a, Reference Maryami, Arcondoulis, Liu and Liu2023c; Maryami & Ali Reference Maryami and Ali2023). The wavelet transform is defined as follows:

(5.2)\begin{equation} W(a,\tau)=\frac1{\sqrt{C_{\psi}}}{\int_{-\infty}^{+\infty} c_{1}(t)\psi^{*}_{a,\tau}{\left(\frac{t-\tau}a\right)}\,\mathrm{d}t}, \end{equation}

where $a$ is the scale dilation parameter corresponding to the wavelet width, $\psi (t)$ is the mother wavelet, $\psi ^{*}((t-\tau )/a)$ is the complex conjugate of the dilated and translated $\psi (t)$, and $C^{-0.5}_{\psi }$ is a constant that takes the mean value of $\psi (t)$.

Figure 25 presents the autocorrelation of the near-field pressure measured at $\theta =90^{\circ }$ and far-field pressure, i.e. $R_{p_{_s}p_{_{s}}}$ and $R_{pp}$, respectively, as a function of normalised time delay, $\tau ^{*}=\tau U_{\infty }/D$, along with time-frequency characteristics of $c_{1}$ extracted from the real part of the wavelet coefficient, ${\rm I\!R}(W)$. For the baseline, both $R_{p_{_s}p_{_{s}}}$ and $R_{pp}$ reveal oscillations within $-20<\tau ^{*}<20$ with a time delay $\tau _{1}^{*}\approx 5.2$, which is associated with the $f_{1}$-tone frequency of $St\approx 0.19$. In this case, ${\rm I\!R}(W)$ exhibits amplitude modulation in time with multiple repetitive patches around the $f_{1}$-tone frequency, which indicates that the turbulent field is a superposition of coherent flow structures (Farge Reference Farge1992). It is observed that $R_{p_{_s}p_{_{s}}}$ presents a strong oscillation with a weak decay rate (slow decrease of peak and trough amplitudes with respect to $\tau ^{*}$) in comparison with $R_{pp}$, revealing the attenuation of sound pressure over a long distance before reaching the far field.

For LB90, while ${\rm I\!R}(W)$ exhibits the amplitude modulation at almost the same energy level as the baseline, the periodic behaviour of $R_{p_{_s}p_{_{s}}}$ and $R_{pp}$ presents a strong oscillation with a decay rate slower than that of the baseline. This confirms that the proximity of the vortex shedding to the cylinder contributes more to strong surface pressure fluctuations and sound radiation compared with the energy content of dominant coherent flow structures. This interpretation receives further evidence by investigating $R_{p_{_s}p_{_{s}}}$, $R_{pp}$ and ${\rm I\!R}(W)$ in the case of LB131. The ${\rm I\!R}(W)$ for LB131 except in ranges of $t\approx 0.2$–$0.5$, 0.6–0.8 and 1–1.3 s reveals amplitude modulation with energy level similar to the baseline, but $R_{p_{_s}p_{_{s}}}$ and $R_{pp }$ are much weaker in oscillation and decay faster compared with the baseline, which is ascribed to a delay in vortex shedding (larger vortex formation size).

5.2. Flow field: contributions from the POD modes

The spatial scale of the dominant coherent structures in the wake, such as Kármán vortices, provides an indication of the overall intensity of the wake fluctuations. These coherent structures can be characterised by their size, which corresponds to the width of the wake (Qu et al. Reference Qu, Wang, Sun, Feng, Pan, Gao and He2017; Duan & Wang Reference Duan and Wang2021). Consistent with previous findings (van Oudheusden et al. Reference van Oudheusden, Scarano, van Hinsberg and Watt2005; Duan & Wang Reference Duan and Wang2021), the first two POD modes for all cases exhibit an orthogonal relationship, as depicted in figures 26 and 27. This characteristic is evident by the coherent structures appearing with a phase shift of ${\rm \pi} /2$ relative to the convection behaviour. Mode 3 exhibits turbulent structures; however, the fluctuation content is weaker compared with modes 1 and 2. The size of the coherent structures gradually expands both laterally and longitudinally as they move downstream, which is in line with the widening of the wake width (refer to figure 11) and the increased convection velocity of the vortices (Zhou & Antonia Reference Zhou and Antonia1992).

Figure 26. Flow structures corresponding to the first three POD modes ($n=1$–3) of the $x$-component of the velocity fluctuations ($u'/U_{\infty }$): (a–d) $n=1$; (e–h) $n=2$; (i–l) $n=3$. The solid circle, square and triangle symbols represent $x\approx L_{vc}$, $L_{r}$ and $L_{f}$, respectively, for each case (see figure 9).

Figure 27. Flow structures corresponding to the first three POD modes ($n=1$–3) of the $y$-component of the velocity fluctuations ($v'/U_{\infty }$): (a–d) $n=1$; (e–h) $n=2$; (i–l) $n=3$. The solid circle, square and triangle symbols represent $x\approx L_{vc}$, $L_{r}$ and $L_{f}$, respectively, for each case (see figure 9).

The strength of the coherent structures diminishes and their energy dissipates in the direction of the mean flow, aligning with the decrease in $\mathrm {TKE}^{*}$ observed in figure 15. This behaviour is consistent with the findings of Portela et al. (Reference Portela, Papadakis and Vassilicos2020) and Hussain (Reference Hussain1983), who demonstrated that these structures dominate the TKE in the near wake and their contribution decreases at a faster rate than the TKE associated with small-scale turbulent structures in the flow regions farther downstream. It is observed that LB41 exhibits significantly lower energy content in its coherent structures compared with the baseline, which can be attributed to its larger vortex formation size. Consequently, these structures do not contribute much to the TKE near the cylinder, consistent with the observations in figure 15(b). On the other hand, for LB90, the coherent structures have a similar energy content to the baseline but appear closer to the cylinder due to the accelerated rolling up of the shear layers. In the case of LB131, the energy level of the coherent structures is also comparable to the baseline, but due to the stretching of the shear layers, these structures are pushed farther downstream. These observations provide further evidence that the proximity of vortex shedding to the cylinder or the size of the vortex-formation region plays a crucial role in attenuating TKE near the cylinder, as well as near-field pressure and acoustic noise. Furthermore, it is evident that each LB case undergoes a specific vortex-shedding mechanism. This is particularly apparent when comparing LB41 and LB131, where both cases result in a larger vortex formation region compared with the baseline, but the energy content of the coherent structures differs significantly.

5.3. Instantaneous flow field: contributions from the POD modes

The time history of the first POD mode time coefficient, $c_{1}$, of the baseline normalised using its corresponding eigenvalue, $\lambda _{1}$, is presented in figure 28(a) in terms of the normalised time, $t^{*}=tU_{\infty }/D$, to investigate the contributions of coherent flow structures to vortex shedding. The time history is characterised by oscillation at the $f_{1}$-tone frequency with small-amplitude modulation. The $t_{0}^{*}-t_{8}^{*}$ labels indicate the instants of two sets of consecutive vortex shedding. The peak-to-peak distance between two successive minima or maxima of the turbulent fluctuation of $c_{1}/\sqrt {2\lambda _{1}}$ within $t_{0}^{*}$–$t_{8}^{*}$ relates to the $f_{1}$-tone frequency.

Figure 28. (a) Time history of the first POD mode time coefficient, $c_{1}$, of the baseline, normalised using its corresponding eigenvalue, $\lambda _{1}$. The red square markers superimposed on the $c_{1}/\sqrt {2\lambda _{1}}$-curve correspond to the instances of two sets of consecutive vortex shedding. The instants are presented using the normalised time, $t^{*}=tU_{\infty }/D$, from $t_{0}^{*}$ to $t_{8}^{*}$. (b,d) Isosurface of normalised instantaneous vorticity, $\omega '^{*}=-0.7$, coloured by $\bar {u}/U_{\infty }$. The black dashed lines represent $x\approx L_{vc}$ and $L_{f}$, and the dashed grey lines illustrate the movement trajectory of the vortices within $a_{0}$–$a{_4}$ and $b_{0}$–$b{_4}$. (c) Flow fields of $\omega '^{*}$ for consecutive instants within $t_{1}^{*}$–$t_{7}^{*}$. The $x$-positions of the vortices at each instant of the first and second sets of shedding are denoted by $a_{0}$–$a{_4}$ and $b_{0}$–$b{_4}$, respectively. These positions are extracted from (c) and figure 9 or table 3.

The normalised instantaneous vorticity, $\omega '^{*}=\omega '_{z}D/U_{\infty }$, in the instant range of $t_{1}^{*}$–$t_{7}^{*}$ is displayed in figure 28(c), where the shear layers are visible on both sides of the cylinder and roll up into vortices in the wake. At $t_{1}^{*}\approx 71$, the shear layer coming from the top side of the cylinder entrains into the growing shear layer of the other side of the cylinder. This entrainment process continues for a quarter of the main tone period up to $t_{2}^{*}\approx 72.5$, corresponding to $c_{1}/\sqrt {2\lambda _{1}}=0$. While not shown in figure 28(c) for brevity, the entrainment process actually begins at $t_{0}^{*}\approx 70$ and becomes strong within $t_{1}^{*}$–$t_{2}^{*}$. At the beginning of the second quarter ($t_{2}^{*}$), the concentrated vortices in the tail of the longer shear layer separate from the shear layer and convect downstream into a vortex. During the second quarter up to instant $t_{3}^{*}\approx 74$, a vortex with vorticity of opposite sign subsequently develops by circulation from the shear layer on the top side of the cylinder until it becomes strong enough to draw the fluid from the other shear layer across the wake. The entrainment of the shear layer coming from the bottom side of the cylinder continues up to $t_{4}^{*}\approx 75$, when the third quarter ends, and then the vortex detaches from the shear layer to facilitate the vortex growth in the fourth quarter for the next vortex shedding. The corresponding flow field at instant $t_{5}^{*}$ appears similar to instant $t_{1}^{*}$ in terms of the vortex shedding process. At instant $t_{7}^{*}$, the shed vortex continues travelling downstream in a behaviour similar to that at instant $t_{3}^{*}$.

The instantaneous flow field in figure 28(c) clearly shows that the first POD mode is ascribed to the periodic vortex shedding at the $f_{1}$-tone frequency. Furthermore, the vortex with high energy content is being shed from the cylinder at the stage, when the entrainment of fluid-bearing vorticity from the other shear layer cuts off the further supply of circulation to the vortex, which then ceases to increase in strength. In line with the findings in figures 26 and 27 and the results presented by Portela et al. (Reference Portela, Papadakis and Vassilicos2020), this implies that the vortex becomes weaker in terms of the energy content after shedding. It is observed in figure 28(c) that the vortex tends to separate from the shear layer at $x\approx L_{f}$.

The time–space evolution of the vortices with a negative vorticity sign fed by circulation from the shear layer on the top side of the cylinder in the instant range of $t_{0}^{*}$–$t_{8}^{*}$ is presented in figure 28(b,d). The vorticity is coloured by $\bar {u}/U_{\infty }$. The location of vortices corresponding to each instant is approximately marked by $a_{0}$–$a_{4}$ and $b_{0}$–$b_{4}$ for the first and second sets of vortex shedding, respectively. The travelling trajectory of the vortices between these locations is presented by dashed grey lines and their slope can be an estimate of the convective velocity of the vortices ($U_{c}$). Interestingly, the velocity magnitude $U_{c}$ in the vortex-formation region between $a_{0}$ ($x/D\approx 0.75$, $t^{*}\approx 70$) and $a_{2}$ ($x/D\approx 1.5$, $t^{*}\approx 72.8$) is approximately four tenths of that outside the vortex-formation region between $a_{2}$ and $a_{4}$ ($x/D\approx 3$, $t^{*}\approx 75$), with values of $U_{c}\approx 0.27U_{\infty }$ and $0.68U_{\infty }$, respectively. This confirms that the entrainment process decreases the $U_{c}$ of the vortices and the opposite takes place after shedding, which is consistent with Zhou & Antonia (Reference Zhou and Antonia1992). From figure 28(b,d), it can be seen that the evolution of vorticity in the shear layer is accompanied by the development of the FAA up to $x\approx L_{vc}$, which is consistent with the observations in figure 8(a).

The evolution of the vortices with a negative vorticity sign in time–space domains for the LB cases is presented in figure 29. It is observed for each LB case except LB90 that the circulation from the shear layer originating from the flow separation on the top side of the cylinder continues to shed vortices farther downstream in comparison with the baseline. This is confirmed by the time–space evolution of $\omega '^{*}$ within $t_{1}^{*}$–$t_{4}^{*}$ as presented in figure 30. It is evident that, for LB41 and LB90, initially, the shear layer from the bottom side of the cylinder is entrained by the one coming from the top side of the cylinder within this time range. This behaviour is opposite to that observed for the baseline (see figure 28c) and LB131. This indicates that vortex shedding in LB41 and LB90 occurs with a noticeable phase shift compared with LB131.

Figure 29. Time history of the first POD mode time coefficient normalised by its eigenvalue, $c_{1}/\sqrt {2\lambda _{1}}$, and isosurface of $\omega '^{*}=-0.7$ coloured by $\bar {u}/U_{\infty }$. The red square markers superimposed on the $c_{1}/\sqrt {2\lambda _{1}}$-curve correspond to the instances of two sets of consecutive vortex shedding from $t_{1}^{*}$ to $t_{8}^{*}$. The $x$-positions of the vortices at each instant of the first and second shedding sets are denoted by $a_{1}$–$a{_4}$ and $b_{0}$–$b{_4}$, respectively. These positions are extracted from figure 9 or table 3. The black dashed lines represent $x\approx L_{vc}$ and $L_{f}$, and the dashed grey lines depict the movement trajectory of the vortices within $a_{1}$–$a{_4}$ and $b_{0}$–$b{_4}$. Here (a) LB41; (b) LB90; (c) LB131.

Figure 30. Flow fields of $\omega '^{*}$ at consecutive instants within $t_{1}^{*}$–$t_{4}^{*}$ extracted from the time history of the first POD mode time coefficient (see figure 29): (a–d) LB41; (e–h) LB90; (i–l) LB131.