## 3 Results
### Grazing Acoustic Signal without Flow  
#### 3.1.1 Self-Designed Experiment Configuration  
Installation of a perforated liner could result in very different overall acoustic damping properties depending on the noise signal incidence angles [28]. In the previous work of the authors [32], the PVPM model was validated for situations where the fluctuating pressure was normal to the perforated plate. In the current paper, in order to validate the PVPM model in the noise signal grazing incidence conditions, an acoustic test rig, as shown in Figures 2 and 3, was constructed. The rig consists of an upstream duct section, a downstream duct section and a liner section installed in between them, each with an internal diameter of 165 mm. Four opposing loudspeakers were installed on the upstream duct. The test rig was designed to include anechoic terminations to minimize reflections from both ends. The pressure measurement uncertainty for those four microphones is within \(\pm\)3%. In this test configuration, a mean flow is not present in the ducts or perforated liners, and other authors' experimental work is cited to validate the PVPM model in the presence of mean flows. Figure 2: Experimental test rig setup. A resonant cavity, Figure 3, was constructed from a 50 \(\times\) 59.4 mm curved rectangular test liner, mounted flush to the side wall of the test section duct, with an attached fan-shaped cavity of 51 mm depth. The volume of the cavity is  
\[\mathrm{V}=\pi[(51\;\mathrm{mm}+165+2\;\mathrm{mm})^{2}-\left((165\;\mathrm{mm}) /2+2\;\mathrm{mm})^{2}\right]\times\frac{41.25^{\circ}}{360^{\circ}}\times 50\; \mathrm{mm}=201\;\mathrm{mL} \tag{16}\]  
where 51 mm is the cavity depth, 165 mm is the internal diameter of the tube, 2 mm is the thickness of the liner and 41.25\({}^{\circ}\) represents the angle of the fan-shaped cavity. The cavity size is typical for distributed Helmholtz resonators installed on real gas turbines in order to attenuate noise within the combustor chamber. A total of 32 circular perforations, as illustrated in Figure 3, were drilled in the curved liners. The diameters of these orifices ranged from 2 to 4 mm. In all cases, the thickness of the perforated plate was fixed at 2 mm. Detailed geometric features of the orifices are listed in Table 1.  
A signal generator and amplifier were employed to generate a 100 dB noise signal within a bandwidth of 100-1000 Hz. Two pairs of microphones were installed in the upstream and downstream ducts, each pair separated by a distance of 10 cm. A four-microphone data processing method [21] was employed to process the data, and the acoustic absorption coefficients within the frequency range 100-1000 Hz were thereby obtained. Numerical simulations, with the PVPM model representing the liner region, were carried out to obtain the equivalent predicted acoustic absorption coefficients. The mesh distribution was as shown in Figure 4, where only two layers of structured meshes are applied to represent the thickness of the perforated liner, which significantly reduce the computational cost otherwise required to resolve the small holes [32]. The sound sources are two loudspeakers mounted perpendicular to the duct walls. With loudspeakers mounted on the side walls, their numerical representation can introduce an unrealistic flow disturbance due to interactions between the additional inlet and the incident grazing mean flow. In this work, the sound source was relocated to its neighboring mean flow inlets/outlet. The area of the acoustic signal inlet represented less than 1/10 of the overall area of the flow boundary so that the original acoustic impedance property of