## 4 Numerical Simulation
### Mechanism Analysis on Noise Reduction  
The time history of the lift coefficient and drag coefficient is compared between the baseline cylinder and wavy cylinder in Figure 18. It can be seen in Figure 18a that, the lift coefficient of the baseline cylinder shows large amplitude fluctuation, ranging from about \(-1.0\) to \(1.0\). The lift coefficient and drag coefficient oscillate with time; this is because the vortex would periodically shed from the rod, leading to the oscillatory nature of the presented coefficients. The oscillatory period in the lift coefficient corresponds to a dimensionless frequency of St = 0.18. It is noted that the oscillatory nature is more obvious for the baseline cylinder than for the wavy cylinder. For the wavy cylinder, the fluctuation of the lift coefficient is greatly reduced, ranging from about \(-0.1\) to \(0.1\). Further analysis shows that, the rms value of the fluctuation of the lift coefficient for the baseline cylinder and wavy cylinder is 0.55 and 0.05, respectively. The fluctuation of the lift coefficient is reduced by as much as 91%. Similarly, Figure 18b shows that the wavy cylinder can also reduce the mean value and fluctuation amplitude of the drag coefficient. The mean value of the drag coefficient for the baseline cylinder and wavy cylinder is 1.36 and 0.98, respectively. The rms value of  
Figure 17: Grid around the baseline cylinder and wavy cylinder. Figure 18: Comparison of the time history of the lift coefficient and drag coefficient between the baseline cylinder and wavy cylinder. the fluctuation of the drag coefficient for the baseline cylinder and wavy cylinder is 0.08 and 0.02, respectively. Overall, the wavy cylinder can not only reduce the drag by 28%, but also reduce the intensity of the drag fluctuation by 75%. The spectrum of the lift and drag coefficient was also compared in Figure 19. The baseline cylinder has obvious characteristic frequency at a Strouhal number 0.18 for the lift coefficient and 0.36 for the drag coefficient. However, the characteristic frequency is no longer so obvious for the wavy cylinder. Moreover, the peak amplitude of the lift coefficient spectrum is reduced by about 24 dB, and the drag coefficient spectrum is reduced by about 35 dB. The rms value of the pressure fluctuation on the cylinder surface is presented in Figure 20 for both the baseline cylinder and wavy cylinder. It can be seen that the pressure fluctuation on the cylinder surface is suppressed very much by the wavy profile. The great reduction in the lift and drag fluctuation intensity, as well as the cylinder surface pressure fluctuation, is beneficial for the noise reduction. Figure 19: Comparison of the lift coefficient and drag coefficient power spectral density. Figure 20: Comparison of the pressure fluctuation on the cylinder surface. Figure 21 shows the iso-surfaces of the Q-criterion for the baseline cylinder and wavy cylinder. The instantaneous vorticity distribution in the vertical plane and different spanwise location is also shown in Figures 22 and 23. It can be seen from Figures 21 and 22 that, the vortex structure for the wavy cylinder shows more complex three-dimensional characteristics. Figure 23 compares the span-wise vorticity distribution for the baseline cylinder and wavy cylinder at three different spanwise locations (trough, middle and peak location). For the baseline cylinder, regular vortex shedding can be observed. For the wavy cylinder, regular vortex shedding can be seen at the peak span location, and less obvious at the middle span location. At the trough span location, the vortex shedding tends to be banded instead of alternate oscitation. This is beneficial to suppress the unsteady load on the cylinder and the resulting noise radiation. The turbulence intensity was also compared between the baseline cylinder and wavy cylinder at three different spanwise locations, as shown in Figure 24. For the baseline cylinder, a large zone of high turbulence intensity can be found in the cylinder near wake.