## 3 Results
### Grazing Acoustic Signal with Bias Flow  
Experimental data by [PERSON] and [PERSON] [21] are hereby cited to validate the PVPM model for the case of a grazing acoustic signal in the presence of a bias flow. Figure 6 illustrates the reported experimental configuration. Two 800 mm-long flow ducts were separated by a 178 mm-long double-layered perforated liner. Both liners were 3 mm thick. The outer liner was a 152 mm-diameter annular duct shell with 2% of its area perpendicularly perforated by 2.7 mm-diameter circular holes. The 127 mm-diameter inner liner was perforated by 0.75 mm-diameter circular holes, and the overall surface porosity was 4%. Those perforations on the inner liner were tilted 45\({}^{\circ}\) toward the downstream duct. The axial region surrounding the two liners formed an inner air cavity. A bias flow was supplied to the cavity and the resultant mean flow with the duct exhausted to a large plenum at the exit of the downstream duct, as shown in Figure 6. This plenum was included in the computational domain in the following numerical simulations. Four loudspeakers were mounted at the end of the upstream flow duct, providing an acoustic signal ranging between 100 and 700 Hz. Transient pressure data were collected and processed by the four-microphone transfer function method as reported in [PERSON] and Dowling [21]. The corresponding numerical simulations were carried out as part of this work, with the PVPM model representing the liner region, to obtain numerical results for acoustic absorption coefficients. The grid resolution is similar to that shown in Figure 4.  
Absorption coefficients derived from the numerical simulations of this work, the experiments of [PERSON] and [PERSON] [21] and also their derived model [21] are plotted against wave numbers at two different bias flow velocities in Figures 7 and 8. M\({}_{\text{in}}\) stands  
Figure 5: CFD and experimental comparison of absorption coefficients in sound grazing incidence conditions with closed cavity. Figure 6: Test rig setup in the experiment by [PERSON] and [PERSON] [21]. for the bias flow Mach number through the inner liner, k is the wave number of the noise signal and L is the length of the liner which is 178 mm. The results from the numerical simulations show very close agreement with both the experimental and model results. The variation in the absorption coefficient with the wave number is successfully captured by the PVPM model. This demonstrates that the combined CFD/PVPM model is able to successfully represent acoustic reflections from the downstream plenum. Despite the overall good agreement, differences of up to 11% do exist between the absorption coefficient curves acquired by the different methods, especially at very high frequencies when kL > 1.7. [PERSON] and [PERSON] [21] suggested that this error may be a result of the high sensitivity of high-frequency signals to the interference between left- and right-traveling waves in the liner section. Figure 8: Numerical and experimental comparison of absorption coefficients at \(\text{M}_{\text{in}}=0.041\) and zero grazing flow conditions [21]. Figure 7: Numerical and experimental comparison of absorption coefficients at \(\text{M}_{\text{in}}=0.023\) and zero grazing flow conditions [21]. Figure 9 demonstrates how acoustic attenuation varies with the bias flow Mach number. All three methods suggest that the optimum absorption effect takes place between \(0.01<\mathrm{M_{in}}<0.02\), and all three methods indicate that a further increase in the bias flow speed above the optimum speed reduces the peak absorption coefficients. On the other hand, the minimum noise damping effect at frequencies \(\mathrm{kL}=1.22\) is hardly changed by the bias flow speed.