## 2 Methods
### Further Development of PVPM Model
#### 2.2.2 Extra Porous Media Region Thickness Corrections Due to Grazing Flows  
The effective acoustic thickness of a perforated liner is larger than the actual physical thickness due to the acoustic radiation effect at the end of the individual perforations. However, the original porous media model ignored such geometry details, and, as a result, the additional thickness due to the end radiation effect is lost. Therefore, when using the PVPM model in a numerical simulation, the thickness of the porous media region is required to be the effective acoustic thickness of the liner rather than the real liner thickness.  
In the presence of significant acoustic pressure fluctuations and strong bias flows, the acoustic end correction length of a perforation is reduced [32]. A number of experiments found that the acoustic end correction length is reduced by the presence of grazing flows.  
[PERSON] [41] proposed a correction factor of \(1/\left(1+305\text{M}_{\text{g}}^{3}\right)\) for end radiation length correction due to grazing flows. However, this term significantly underestimates the effect of grazing flows according to some other researchers [23, 43]. [PERSON] and [PERSON] [43] introduced a grazing flow acoustic reactance term, \(-0.3\text{M}_{\text{g}}/\sigma\). However, experiments by [PERSON], [PERSON] and [PERSON] et al. [44, 45] clearly showed that the acoustic reactance did not decrease linearly with the grazing flow Mach number, as suggested. According to [PERSON] and [PERSON] [24], the end correction length starts to decline within moderate grazing flows, and the decrease in the end correction length slows down with increasing grazing flow speed. [PERSON] et al. [45] found the same trend where the decrease in acoustic reactance with grazing flow speeds nearly stopped if \(\text{M}_{\text{g}}>0.15\).  
Grazing flows reduce the effect of acoustic radiation, as represented by the end correction length, in two different ways. First, a grazing flow-induced local neck flow reduces the end correction length in the same way as with bias flows, as described in Equation (10). Second, a grazing flow \"blows away\" the acoustic radiation on that side of the liner. The correction factor due to the \"blowing away effect\" is slightly under 0.5 when the high-speed grazing flow (\(\text{M}_{\text{g}}>0.15\)) blows away all acoustic radiations at the flow side and also some air mass in the internal rim of oifices. Therefore, on the basis of Equation (8), the effective thickness of perforated liners is thus proposed to be  
\[l_{\text{eff}}=l_{\text{t}}+0.85\text{d}\big{(}1-1.4\sqrt{\sigma}\big{)}/ \left\{\begin{array}{c}1+\frac{\text{M}_{\text{b}}}{\sigma}\quad\quad if \text{M}_{\text{g}}=0\\ \big{(}1+10\text{M}_{\text{g}}\big{)}\big{(}1+\frac{\text{M}_{\text{b}}}{ \sigma}\big{)},\quad if\text{M}_{\text{b}}/\text{M}_{\text{g}}>C_{\text{g}}\\ \big{(}1+10\text{M}_{\text{g}}\big{)}\big{(}1+\frac{\text{c}_{\text{g}}M_{ \text{g}}}{\sigma}\big{)},\,if\text{M}_{\text{b}}/\text{M}_{\text{g}}<C_{\text {g}}\end{array}\right. \tag{15}\]  
The term \(\big{(}1+10\text{M}_{\text{g}}\big{)}\) considers the \"blowing away\" effect by the grazing flow. This term is no greater than 2.5 because the \"blowing away\" effect takes place only at the flow side of the liner and does not affect the acoustic radiation at the other side. The second term \(\big{(}1+C_{\text{g}}\text{M}_{\text{g}}/\sigma\big{)}\) is to reflect the local neck flow effect, and this term will be replaced by \(1+\text{M}_{\text{b}}/\sigma\) if the inertial resistance caused by bias flows is stronger than that due to grazing flows [45].