# Prediction of the Absorption Characteristics of Non-Uniform Acoustic Absorbers with Grazing Flow
## 4 Absorption Characteristics of the Duct-Acoustic Absorber System
### Rectangular Duct Acoustic Absorber  
As shown in Figure 4, the cavity length \(h\) dominated the imaginary part of the overall impedance and hence determined the resonance frequency. Thus, the basic resonant elements with different cavity lengths can be combined together to generate the desired resonance frequency distribution and expand the absorption bandwidth. To this end, a broadband absorption unit (BAU) absorber consisting of six BREs with the same cavity thickness \(a=10\) mm and different cavity lengths was constructed, as shown in Figure 14. The six cavity lengths are \(h_{1}=28\) mm, \(h_{2}=21\) mm, \(h_{3}=20\) mm, \(h_{4}=19\) mm, \(h_{5}=18\) mm and \(h_{6}=11\) mm. They were determined by the maximum average transmission loss of the BAU absorber with a large number of different cavity lengths. The hole diameter \(d=0.8\) mm and perforation rate \(\sigma=8.04\%\) were used in the following simulations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & **Average Error** & **Peak Frequency Error** & **Peak Value Error** \\ \hline IBNSE & 5.87\% & 1.75\% & 4.69\% \\ FNSE & 4.24\% & 1.67\% & 2.50\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Transmission loss errors corresponding to the FNSE and the IBNSE methods.  
Figure 13: Comparisons of transmission loss and absorption coefficient of the structure in Reference [15]. (**a**) Transmission loss and (**b**) acoustic absorption coefficient.  
For a practical design of a duct acoustic absorber system, the acoustically treated area in the duct should be large enough, which can be achieved by increasing the number of BAUs. To this end, a planar non-uniform acoustic absorber consisting of 20 BAUs was constructed, as shown in Figure 15.  
Referring to Figure 1, the designed planar non-uniform acoustic absorber was placed in the rectangular duct as an acoustically treated liner. In simulations, the length and width of the duct section were \(A_{1}=B_{1}=68\) mm, the total length of the duct was \(L_{\text{duct}}=810\) mm and the length of the planar acoustic absorber was \(L=410\) mm. The incident acoustic pressure level was taken as 120 dB. Again, for comparison purposes, both the FNSE and the IBNSE methods were used in simulations. Figures 16 and 17 show the geometric and FE models of the rectangular duct-planar acoustic absorber (RDPAA) system suitable for the FNSE method and IBNSE method, respectively.  
Figure 18 reveals that under different Mach numbers, the transmission loss and absorption coefficient obtained by the IBNSE method agreed very well with those obtained by the FNSE method, which once again proves the correctness of the IBNSE method. The FNSE method consumed 10 h and 20 min per calculation (corresponding to one Mach number), while the IBNSE method consumed only 3 h and 5 min per calculation, which demonstrates that the developed IBNSE method is computationally more efficient. It was observed that with the increase of incidence frequency, the transmission loss and acoustic absorption coefficient first increased and then decreased. The reason for this phenomenon can be explained by Figure 2h. As the frequency increases, the imaginary part of impedance of the perforated plate increases, which makes the imaginary part of the overall impedance change from negative to positive, with the point at which the imaginary part of the impedance is zero corresponds to the absorption peak. In addition, with the increase of Mach number, the peak values of transmission loss and absorption coefficient decreased, which led to a more uniform absorption performance within the considered frequency range. The reason is that an increase of Mach number leads to an increase in acoustic resistance and a decrease in acoustic reactance. The increased acoustic resistance usually results in a decreased peak value and increased bandwidth, while the decreased acoustic reactance shifts the absorption peak to a higher frequency.