## 3 Results and Discussions
### Coupled Flow and Noise Simulations for a Wind Turbine Array
#### 3.4.4 Wind Turbine ANPropagation across Multi-Wake
Repeating the calculation for many frequencies and extracting the data at a receiver 1500 m away and 2 m height, a full spectrum of \(\Delta L\) will be obtained. Using Equation (12) and considering all the effects (including \(\Delta L\)), the SPL at receiver's location under Case 1 and Case 3 is gathered and shown in Figure 15. Besides, the spectrum of ANSource and at receiver's location (without \(\Delta L\)) are also depicted in Figure 15. In this work, the ANSource is considered in an \(1/3\)-octave band from 20 Hz to 5000 Hz. Since a minimal eight grid points are needed to resolve a sound wave, the grid number for high-frequency simulations increases sharply if the same domain size is adopted. For instance, there is a 100-fold increase in the grid number for frequency of 3000 Hz compared with that of 300 Hz. To reduce the computational effort, frequencies above 5000 Hz are not considered here. Another reason is that the noise reduction due to air absorption, \(-\alpha D\) in Equation (12), is much larger than the other factors at large frequency (\(f\) > 5000 Hz). The SPL at the source location is shown with black line with asterisk and labeled as 'Source: Lp' in Figure 15. The black lines with stars represent the SPL at receiver (2 m height at \(x\) = 1500 m) without considering the propagation loss \(\Delta\)L in Equation (12). The blue line (Case 1) and red line (Case 3) consider all the factors, including geometric attenuation, air absorption and \(\Delta L\), in the long-path ANPropagation simulations. The big deviation between the two black lines demonstrates that the geometrical attenuation (\(10\log_{10}4\pi D^{2}\)) contributes most to the ANPropagation loss. The even larger reduction of noise, when \(f\) > 1000 Hz, means that air absorption has a larger effect at high frequencies. The differences between blue/red lines and the black line with stars implies the value of \(\Delta\)L. The value of blue and red lines are larger than that of black line with  
Figure 14: The RSPL at 2 m height extracted from four cases of Figure 13.  
stars, which means positive \(\Delta\)L near low frequency (\(f\) = 30 Hz). However, Case 1 and Case 3 show opposite \(\Delta\)L around \(f\) = 200 Hz which implies the influences of wake and downward refraction. Similar as the previous simulations, the following study focuses on the noise propagation area between 1120 m and 2500 m, and a higher frequency \(f\) = 1000 Hz is selected. In this region, the three individual wind turbine noise sources propagate and superimpose on each other. In Figure 16a, the results cover the entire domain behind the last wind turbine. The solution differs from Figure 13a in the sense that it is the RSPL from all the three wind turbines. In Figure 16b, the uniform sound profile at the inlet boundary is affected by the multi-wake (see Figure 12a), such that the resulted RSPL is enhanced around wake center. With the prescribed sound profile of Figure 12b, the RSPL is refracted downwards, as shown in Figure 16c. Similarly, using the sound profile given in Figure 12c, the resultant sound propagation is refracted upwards, see Figure 16d. Figure 17 better illustrates the superposition of the sound pressure propagates from each wind turbine, which is a decomposition of Figure 16d. In the first figure, noise is generated from wind turbine #1 and propagates through the wake flow. The other two wind turbines are depicted with white lines, which imply that they are currently excluded for noise propagation. In the next simulation, shown in Figure 17b, ANPropagation from wind turbine #2 is individually considered, which excluded noise propagation from wind turbine #1 and #3. Similarly, wind turbine #3 is then considered for ANPropagation, which does not take account of the two upstream wind turbine noise propagations. The total RSPL is the summation of sound pressure following the logarithmic sumation rule, which is shown in Figure 17d. Figure 18 shows the RSPL or \(\Delta L\) extracted from a line at 2 m high.