## 2 Methods
### Method Validation  
In order to verify the accuracy of the proposed approach, a straight-through perforated tube silencer with two different perforated patterns that was proposed by the authors of [(20; 21)] and a cross-flow perforated tube silencer that was proposed by the authors of [(10)] was considered. Previously, [PERSON] et al. [(22)] employed the 3D time-domain CFD approach to calculate the _TL_ of two perforated tube silencers at different flow velocities and temperatures, and their predictions were verified by experimental data. Moreover, they found that the distribution of flow velocity inside the silencer was anisotropic and nonhomogeneous. Therefore, it is not sufficiently accurate to consider the actual gas flow as the mean flow. The approach proposed in this paper avoids this problem effectively.  
Figure 2 presents the straight-through perforated tube silencer. The diameters of the inner and outer cavities are \(d\) = 32 mm and \(D\) = 110 mm, respectively; the length of the silencer is \(l\) = 200 mm; the thickness of the wall is 2 mm; and the diameter of the hole and the porosityare \(dh\) = 4 mm and \(\sigma\) = 4.7%, respectively, for Pattern 1, and \(dh\) = 8 mm and \(\sigma\) = 14.7%, respectively, for Pattern 2. The cross-flow perforated silencer is illustrated in Figure 3. The diameters of the inner and outer cavities are \(d\) = 49.3 mm and \(D\) = 101.6 mm, respectively; the lengths of tubes on both sides of the baffle are \(l_{1}\) = \(l_{2}\) = 128.6 mm; each tube is perforated with 160 orifices, with a porosity of 3.9%, a diameter of 2.49 mm; and a wall thickness of 0.81 mm.  
Figure 1: Sketch of data transfer (Number of nodes \(N\) = 3, Maximum distance \(R\) = 100 mm).  
Figure 2: Straight-through perforated tube silencer.  
The proposed approach is applied to calculate the \(TL\) for the silencers under different boundary conditions, and the predictions were compared with the available measurement results. Figure 4a and 4b compare the \(TL\) curves of the straight-through perforated tube silencer for Pattern 1 and Pattern 2, respectively. Considering the predictions and measurements at an air flow of \(M=0.1\) and a temperature of \(T=288\) K, it is evident that the predictions are in line with the experimental data. The predicted and measured TL curves for Pattern 1, with \(M=0.2\) and \(T=288\) K, are presented in Figure 5, and it is evident that these predictions are also in line with the experimental data. Figure 6 presents a comparison of the numerical results and the measurements for the cross-flow silencer with an air flow velocity of \(v=17\) m/s and \(T=347\) K, which again presents an excellent agreement.  
Based on the above comparisons, it is evident that the proposed approach displayed a high degree of accuracy in investigating the acoustic attenuation performance of perforated tube silencers. This high degree of accuracy should be attributed to the avoidance of using simplified temperature fields and air flow in the calculation process. In doing so, the actual working condition of the silencer can be  
Figure 4: Comparison of the predicted and measured transmission loss (\(TL\)) for the straight-through perforated tube silencer (Mach number \(M=0.1\), Temperature \(T=288\) K): (**a**) Pattern 1; (**b**) Pattern 2.  
Figure 3: Cross-flow perforated tube silencer.  
better reflected. Additionally, it must be stated that the computations may be highly time-consuming because there are too many orifices in the silencers, which increases the amount of mesh.