## 2 Experimental Setup
### Noise Measurement  
The far-field noise measurement has been conducted using an arc microphone array with nine microphones, depicted in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cylinder No.** & \(D_{m}\) **(mm)** & \(\lambda\) **(mm)** & \(a\) **(mm)** & \(r=\lambda/a\)** \\ \hline \#0 & 22 & \(\infty\) & 0.01 & \(\infty\) \\ \#1 & 22 & 132 & 3.3 & 40 \\ \#2 & 22 & 132 & 4.4 & 30 \\ \#3 & 22 & 132 & 5.5 & 24 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of wave profile on all the cylinders.  
Figure 4: Installation of the experimental model in the test section.  
Figure 3: Schematic map of cylinders with different aspect ratios.  
The center of the arc was set to the axis of the cylinder, the interval between the adjacent microphones was \(8^{\circ}\), and the radius of the circular arc was 2.05 m. Thus, the microphones are located at about 2.05 m from the cylinder, which presents the circular antenna to capture the far-field noise characteristics of the cylinder. The uncertainty of the microphone is \(\pm\)0.2 dB. Each microphone has been calibrated to satisfy the experimental requirements. The sampling frequency is 51.2 KHz with the sampling time to be 10 s. The far-field acoustics radiation is measured for the cylinder under four airflow speeds (U = 30, 40, 50, and 60 m/s). The Reynolds number is respectively 4.4 \(\times\) 10\({}^{4}\), 5.9 \(\times\) 10\({}^{4}\), 7.4 \(\times\) 10\({}^{4}\) and 8.8 \(\times\) 10\({}^{4}\).  
SPL (sound pressure level) is defined as follows [52]:  
\[SPL=10\log_{10}\left(\phi(\theta_{i})/P_{0}^{2}\right) \tag{2}\]  
\[dSPL=SPL_{i}-SPL_{i0} \tag{3}\]  
where \(\phi(\theta_{i})\) is the pressure spectrum density at the \(i\)th microphone and \(P_{0}\) is the reference sound pressure. \(SPL_{i}\) and \(SPL_{i0}\) are the sound pressure level of the wavy cylinder and baseline cylinder at the \(i\)th microphone, respectively. \(dSPL\) is the increment between \(SPL_{i}\) and \(SPL_{i0}\).  
The cylinder model can be regarded as a linear sound source due to cylindrical far-field acoustic radiation. Thus, the sound power and \(PWL\) were defined in the following formulas:  
\[\left\{\begin{array}{l}W=\frac{L}{\rho c_{0}}\sum\limits_{i=1}^{i=N}\left( \phi(\theta_{i})\cdot R_{i}\cdot\Delta\theta_{i}\right)\\ PWL=10\log_{10}(W/W_{0})\end{array}\right. \tag{4}\]  
where \(W\) is the sound power integrated between the radiation angles 221\({}^{\circ}\) to 285\({}^{\circ}\). \(L\) is the span length of cylinder. \(\rho\) is the air density and \(c_{0}\) is the sound speed. \(R_{i}\) is the distance between the axis of the cylinder and the \(i\)th microphone. \(\theta_{i}\) is the included angle between airflow direction and the ith microphone. \(\Delta\theta_{i}\) is the angle between adjacent microphones. \(W_{0}\) is the reference sound power.  
_OASPL_ (Overall Sound Pressure Level) is defined as follows [54; 55]:  
\[E_{i}=\int_{f_{\min}}^{f_{\max}}\phi(i)\cdot df \tag{5}\]  
\[OASPL_{i}=10\log_{10}\left(E_{i}/P_{0}^{2}\right) \tag{6}\]  
Figure 5: Schematic and top view of the set-up of the arc microphone array.  
_DOASPL_ (Difference in Overall Sound Pressure Level) is the overall sound pressure level difference between the baseline cylinder and wavy cylinder as follows:  
\[DOASPL=\frac{\sum\limits_{i=1}^{i=N}(OASPL_{i0}-OASPL_{i})}{N} \tag{7}\]