# Prediction of the Absorption Characteristics of Non-Uniform Acoustic Absorbers with Grazing Flow
## 3 Impedance Boundary Navier-Stokes Equations (IBNSE) Method
### IBNSE Method Using Semi-Empirical Impedance Model
Next, the influences of the MPP parameters on the acoustic attenuation performance were studied using the IBNSE method with Goodrich model and the FNSE method. In simulations, let the hole diameter \(d\) and the perforation rate \(\sigma\) in the four basic parameters given in Section 3.2 be the variable parameters, and the other three parameters are fixed. In Figure 8a, three MPP cases with hole diameters of 0.64, 0.8 and 1.06 mm were considered. Correspondingly, the number of holes \(n\) was changed to 25, 16 and 9, so that the perforation rate \(\sigma\) remains unchanged at 8.04%. In Figure 8b, three MPP cases with perforation rates of 4.52%, 8.04% and 12.57% were used, and the number of holes \(n\) was changed accordingly to 9, 16 and 25, so that the hole diameter \(d\) remains unchanged at 0.8 mm. \begin{table}
\begin{tabular}{c c c c c} \hline
**Impedance** & **Mach Number \(M\)** & **Hole Diameter \(d\)** & **Perforation Rate \(\sigma\)** & **Frequency \(f\)** \\ \hline [PERSON] model & 8.38\% & 4.22\% & 5.65\% & 5.86\% \\ [PERSON] model & 9.68\% & 3.59\% & 3.77\% & 8.86\% \\ [PERSON] model & 14.07\% & 7.15\% & 10.32\% & 11.66\% \\ [PERSON] model & 2.45\% & 1.63\% & 3.22\% & 3.28\% \\ \hline \end{tabular}
\end{table} 
Table 1: Average relative errors of the predicted transmission loss using the IBNSE method under different parameter variations (Mach number, hole diameter, perforation rate and frequency). Figure 7: Comparisons of transmission loss using different impedance models. (**a**) Different Mach number; (**b**) different hole diameters; (**c**) different perforation rates; (**d**) different frequencies. From Figure 8a, it can be seen that a larger hole diameter produced a higher peak of transmission loss, and the peak frequency decreased with the increase in hole diameter. Figure 8b shows that the amplitude and frequency of the transmission loss peak increased with the increase in perforation rate. Within the given frequency band, the IBNSE method yielded accurate results using different perforated plate parameters. In order to further verify the accuracy of the Goodrich model, simulations are carried out under the perforation parameters \(d=0.8\) mm and \(\sigma=8.04\%\). The transmission losses were predicted by the IBNSE method with Goodrich model under different Mach numbers and compared with the results obtained by the FNSE method, as shown in Figure 9. Again, the transmission loss at different Mach numbers obtained by the IBNSE method with Goodrich model were in good agreement with those computed by the FNSE method. In addition, it can be found that the peak value of transmission loss decreased with the increase of the Mach number. This is because the high grazing flow velocity increases the acoustic resistance of the overall structure through \(R_{cm}\) in Equation (21), the total relative surface resistance has a larger difference with respect to 1 and the transmission loss is roughly inversely proportional to the absolute value of the total impedance of the absorber [45].