## 2 Methods
### Further Development of PVPM Model  
#### 2.2.1 Extra Flow Resistance Due to Grazing Flow Effect  
A grazing flow is a flow which travels in parallel with a perforated surface. A grazing flow impacts on the edges of perforations, creating local flow phenomena, as illustrated in Figure 1[35]. The non-uniform flow in the region of the neck of the hole alters the flow resistance of the liner due to inertial losses. However, the extra inertial resistance due to non-uniform neck flows is not directly represented by the homogeneous porous media model. This must be taken into account by incorporating an additional flow resistance in the pressure drop source terms for the porous media region (see Equation (7)). According to a series of experiments [36; 37; 38; 39; 40], the specific flow resistance r\({}_{\text{g}}\) caused by grazing flows is proportional to the grazing flow Mach number M\({}_{\text{g}}\):  
\[\text{r}_{\text{g}}=\frac{\text{C}_{\text{g}}\cdot\text{M}_{\text{g}}}{\sigma} \tag{11}\]  
Figure 1: Illustration of a grazing flow-induced neck flow in a single perforation. The proportional constant \(C_{g}\) is different across different experiments. \(C_{g}\) was claimed to be between 0.3 and 0.5 in these experiments. [PERSON] et al. [40] further confirmed that the coefficient \(C_{g}\) varies with the dimensions of perforations and, according to him,  
\[r_{g}=\frac{M_{g}}{\sigma}/\left(2+1.256\frac{\delta}{d}\right) \tag{12}\]  
where \(\delta\) is the boundary layer thickness displacement due to grazing flow incidence. [PERSON] et al. [37] found that \(\delta\) is approximately 1.45 mm and hardly changes with strong grazing flow speeds. Therefore, grazing flow-induced normalized specific flow resistance of the porous media region is treated in this paper as  
\[r_{g}=\frac{M_{g}}{\sigma}/\left(2+\frac{0.0018}{d}\right) \tag{13}\]  
The constant \(C_{g}=1/\left(2+\frac{0.0018}{d}\right)\) varies from 0.5 for large perforations to 0.25 for small perforations. This variation in the grazing flow stirring effect with orifice sizes is intuitively understandable, where a stronger local neck flow will be created in a larger perforation due to the larger orifice open to impact by the grazing flow. Grazing flows and bias flows coexist in the perforated liner of a gas turbine combustion chamber. A number of experiments [9; 41; 42] reported that the grazing flow is important only when the mean bias flow through the neck is relatively weak. [PERSON] [9] found that the grazing flow and bias flow enhance noise damping effects to the same degree when \(M_{b}/M_{g}=1/3\) and the effect of the grazing flow can be ignored if \(M_{b}/M_{g}>1/3\). Similarly, [PERSON] [41] proposed the ratio \(M_{b}/M_{g}\) to be 0.3, above which the grazing flow effect is negligible. Otherwise, when the grazing flow is much stronger than the bias flow, grazing flow-induced resistance is dominant over bias flow-induced resistance. Both [PERSON] [9] and [PERSON] [41] indicated that the overall flow resistance of a perforated liner is dominant either by the bias flow or the grazing flow, and it is sensible to take the larger resistance to be the resistance of the liner. Bias flow-induced flow resistance, \(r_{b}\), is directly resolved and accounted for by the current PVPM model, as discussed by [PERSON] et al. [32]. The flow resistance due to a bias flow does not need to be imposed in the pressure loss source terms. However, the effect of grazing flows was not included in the original homogenous PVPM model. As a consequence, the overall flow resistance \(r\) of a perforated liner represented by the PVPM model is required to be amended by the grazing flow-induced resistance, as shown in Equation (14):  
\[r=\left\{ \begin{array}{cc}0&\text{if }r_{b}\geq\frac{C_{g}M_{g}}{C_{g}M_{g}} \\ \frac{C_{g}M_{g}}{C_{g}M_{g}}-r_{b}&\text{if }r_{b}<\frac{C_{g}M_{g}}{\sigma} \\ \frac{C_{g}M_{g}}{\sigma}&\text{if }r_{b}=0\end{array} \right.