# Prediction of the Absorption Characteristics of Non-Uniform Acoustic Absorbers with Grazing Flow
## 4 Absorption Characteristics of the Duct-Acoustic Absorber System
### Annular Duct Acoustic Absorber  
In addition to the rectangular duct, the planar BAU can be modified and applied to the annular duct structure. In Figure 19, the absorber with a circumferentially bent BAU is placed on the inner wall of the annular duct to form an annular duct-cylindrical acoustic absorber (ADCAA) system. In Figure 19, the outer and the inner radii of the duct  
Figure 17: Geometric and detailed FE models of the RDPAA system with grazing flow (suitable for the IBNSE method). (**a**) Geometric model and (**b**) FE model. Figure 18: Comparisons of transmission loss and absorption coefficient of the RDPAA system under different grazing flow Mach numbers. The solid lines represent the results computed by the IBNSE method with Goodrich model, while the circles represent the results computed by the FNSE method. (**a**) Transmission loss and (**b**) acoustic absorption coefficient. are \(R_{1}=140.2\) mm and \(R_{2}=110.2\) mm, respectively. The inner radius of the acoustic absorber is \(r=99.4\) mm and the outer radius is equal to \(R_{2}\). The total length of the duct is \(L_{\text{duct}}=740\) mm and the total length of the cylindrical acoustic absorber is \(L=330\) mm. In order to adapt the planar BAU to the cylindrical surface, the perforated plate and the cavities in the planar BAU were bent along the circumference of the duct, while the volume of each cavity was kept unchanged, as shown in Figure 20. The bent BAU structure constitutes the basic acoustic absorption unit of the cylindrical non-uniform acoustic absorber. In Figure 20, the inner radius \(r=99.4\) mm, the central angle of the bended BAU was \(\theta=21.95^{\circ}\) and the axial length was \(l=34\) mm. The central angles corresponding to the six cavities were \(15.37^{\circ}\), \(11.52^{\circ}\), \(10.98^{\circ}\), \(10.43^{\circ}\), \(9.88^{\circ}\) and \(6.04^{\circ}\). The other parameter values were consistent with the original planar BAU structure. The final cylindrical acoustic absorber was constructed by arranging 16 and 10 bent BAUs along the circumferential and axial directions, respectively, as shown in Figure 21. Then, the cylindrical absorber was placed in the inner wall of the annular duct to form an ADCAA system. The FE models that are suitable for the FNSE method and the IBNSE method are shown in Figures 22 and 23, respectively, in which only 1/4 of the cylindrical structure with symmetric bou  
Figure 19: The ADCAA system. Figure 20: The bent BAU as the basic building block of the cylindrical non-uniform acoustic absorber. It can be seen form Figure 24 that under different Mach numbers, the transmission loss and absorption coefficient obtained by the IBNSE method also agreed very well with those obtained by the FNSE computations. The designed cylindrical acoustic absorber had good absorption performance over a very wide frequency range (absorption coefficient was higher than 50%). The absorption characteristics of the ADCAA system were similar to those of the RDPAA structure. However, the influence of Mach number on the absorption coefficient of the ADCAA system was smaller than that of the RDPAA system, which resulted in relatively flat absorption curves for different Mach numbers. This reveals that compared with RDPAA system, the absorption performance of the ADCAA system is not sensitive to the variations of Mach number. In this example, the FNSE method consumed 25 h and 40 min per calculation, while the IBNSE method consumed 8 h and 30 min per calculation, demonstrating that the calculation efficiency was greatly improved. Figure 21: The cylindrical non-uniform acoustic absorber formed by the bent BAUs. Figure 23: The geometric and detailed FE models of the ADCAA system with grazing flow (suitable for the IBNSE method). (**a**) Geometric model and (**b**) FE model. Figure 22: The geometric and detailed FE models of the ADCAA system with grazing flow (suitable for the FNSE method). (**a**) Geometric model and (**b**) FE model.