# Flow-Induced Noise Characteristics Analysis of a Pipeline Structure in a Cabin Rigid Corner
## 3 The Acoustic Radiation Equation of the Pipeline Structure in a Rigid Corner
### Double-Reflection Theory  
According to the research results by [PERSON] [20], [PERSON] [21,22] and [PERSON] [24], the double reflection method is highly effective to solve the acoustic radiation problem in the quarter-infinite acoustic space. The double reflection method states that the quarter-infinite space has four acoustic radiation sources, as shown in Figure 3.  
In the 2D domain, resource \(\mathrm{O_{T}}\) (the solid circle) is the actual cylindrical shell and the cylindrical shell \(O_{1}\) (the dotted ring) and cylindrical shell \(O_{2}\) (the dotted ring) are formed by the rigid wall, with cylindrical shell \(O_{3}\) (the dotted ring) being formed by the double reflection mirrors. Since three images exist, the acoustic pressure \(p(r,\theta,z)\) of the acoustic field point \(\mathrm{P}\) is superimposed by four parts:  
\[p(r,\theta,z)=p_{1}(r_{1},\theta_{1},z)+p_{2}(r_{2},\theta_{2},z)+p_{3}(r_{3}, \theta_{3},z)+p_{T}(r_{T},\theta_{T},z), \tag{7}\]  
Figure 3: The double reflection of the cylindrical shell. (The circular shape in first quadrant is actual cylindrical shell, with circular images in the second, third and fourth quadrants).  
Figure 2: Pipeline structure location. (The left figure shows the position of pipeline structure in the quarter-infinite space. The right figure shows the simplified 2D problem of the pipeline structure).  
the fluid satisfies the no-slip condition in the fluid-solid interface; and the mean velocity of the flow is 2 and 5 m/s, satisfying the continuity condition at the fluid-solid interface. A dense FE mesh is applied to the fluid-solid interface to achieve high numerical accuracy, enhancing the numerical computation precision in the turbulent pulse pressure analysis. The computational domain mesh is shown in Figure 4.  
Numerical analysis of the turbulent pulse pressure in the pipeline was based on LES theory. The Smagorinsky-Lilly sublattice model was applied to capture information regarding the vortex structure inside the pipeline structure. The turbulent pulse pressure was applied to the pipeline as an excitation loading action. In the numerical calculation, the calculation step was \(10^{-6}\) s and the total calculation time was 2 s. The influence of the flow velocity on the turbulent fluctuation pressure inside the pipeline was analyzed. In the turbulent vortex distribution analysis, the \(Q\) criteria were used to evaluate the vortex characteristics and \(Q\) = 10 was chosen in the present work, as shown in Figure 5.  
The turbulent pulse pressure at the inner surface of the pipeline structure was transformed from time domain to frequency domain. Three points previously identified in Figure 4 were used as reference points for the flow field to compare the influence of the flow velocity on the fluctuation pressure, as shown in Figure 6.  
Figure 4: Fluid incoming flow model. (Point A, point B and point C are locations on the inner surface of the pipeline. The distance between point A and point C is L. Point A is the flow field entrance, point C is the flow field outlet, and point B is the midpoint of point A and point C).  
Figure 5: Vortex distribution contour in the pipeline. (The mean flow velocity is 2 m/s in (**a**) and the mean flow velocity is 5 m/s in (**b**)).  
Figure 6: _Cont._  
The turbulent pulse pressure was concentrated in the low-frequency band. With increasing flow velocity, the turbulent fluctuation pressure increased significantly, and the peak value of fluctuation pressure shifted towards the high-frequency direction. The fluctuation pressure at the exit point was significantly greater than at the entrance point, regardless of high flow velocity or low flow velocity.