# 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  
Different from the FNSE method, the developed IBNSE method uses the transfer impedance to replace the perforated plates with an impedance plane. The computational flow chart of the IBNSE method is shown in Figure 3.  
Figure 2: Comparisons of different impedance models. (**a**) Real part of impedance vs. Mach number; (**b**) imaginary part of impedance vs. Mach number; (**c**) real part of impedance vs. hole diameter; (**d**) imaginary part of impedance vs. hole diameter; (**e**) real part of impedance vs. perforation rate; (**f**) imaginary part of impedance vs. perforation rate; (**g**) real part of impedance vs. Frequency; (**h**) imaginary part of impedance vs. frequency. In the IBNSE method, the impedance boundary conditions satisfy the following equations  
\[\boldsymbol{\sigma}_{\text{up}}-\boldsymbol{\sigma}_{\text{down}}=-(p_{\text{up}} -p_{\text{down}})\boldsymbol{n},\ p_{\text{up}}-p_{\text{down}}=-Z_{M}( \boldsymbol{n}\cdot\boldsymbol{u}) \tag{24}\]  
where \(\boldsymbol{\sigma}_{\text{up}}\) and \(\boldsymbol{\sigma}_{\text{down}}\) are the stress tensor of both sides of the perforated plate and \(p_{\text{up}}\) and \(p_{\text{down}}\) are the pressures of both sides of the perforated plate, respectively. Since the detailed modeling and meshing of micro-perforated plates are omitted, the computational efficiency of the IBNSE will be greatly increased. In order to determine which semi-empirical model is the most accurate, a basic resonance element (BRE) was first constructed, which includes an MPP and a backing cavity, as shown in Figure 4. The micro holes reflect acoustic resistance, while the cavity contributes to acoustic reactance, so the combination of MPP and the cavity works as an acoustic absorber. In Figure 4, the width of the MPP is \(a=10\) mm and the length of the MPP and the cavity is \(h\). Note that \(h\) is a variable parameter used to adjust the volume of the cavity. Since the cavity length is extended along the MPP surface, different cavity volumes can be designed to produce different resonance frequencies without increasing the thickness of structure. The thickness of the MPP is \(t\) and all the wall thicknesses of the cavity are \(b=1\) mm. The width and the height (thickness) of the cavity are the same as the width of MPP. The number of micro holes is \(n\), the hole diameter is \(d\) and the perforation rate of MPP is \(\sigma\). The air density is \(\rho_{0}=1.2\) kg/m\({}^{3}\), sound velocity is \(c_{0}=343\) m/s and dynamic viscosity coefficient is \(\mu=1.8\times 10^{5}\) Pa \(\cdot\) s.  
Next, the FNSE and IBNSE methods were used to verify the accuracy of these semi-empirical impedance models. To this end, a basic resonance element (BRE) was placed on the lower wall of the rectangular duct, forming a rectangular duct-BRE (RDBRE) system. Referring to Figure 1, the length and width of the duct section are \(A=B=12\) mm, the total length is \(L_{t}=80\) mm and the length of the cavity \(h\) in Figure 4 was set as 15 mm. In this validation process, the grazing flow Mach number was taken as 0.1 and the incident acoustic pressure level was 120 dB. The acoustic attenuation characteristics of the BRE absorber under grazing flow were computed by the FNSE and IBNSE methods, respectively. In the FNSE simulations, all the detail structures including all micro-holes and backing cavities in the BRE should be modeled and meshed, as shown in Figure 5. Boundary layers were used on the duct wall and hole wall. The number of layers was 6 and the stretch factor of the boundary layers was 1.2. The free tetrahedral grid was used for  
Figure 3: Computational flow chart of the impedance boundary Navier–Stokes equations method. method using the Goodrich model had the highest accuracy for the different parameter variations. Therefore, the Goodrich model was used for calculations in the following IBNSE simulations.