# Structural Design and Parameter Optimization of Bionic Exhaust Tailpipe of Tractors
## 2 Materials and Methods
### Boundary Conditions and Acoustic Solution Settings  
In this research, the simulation software ANSYS Fluent was used to calculate the internal flow field of the exhaust tailpipe to predict the aerodynamic noise generated in the exhaust tailpipe and analyze the noise reduction mechanism of the bionic exhaust tailpipe. First, a large eddy simulation (LES) was used to solve the Navier-Stokes equations, and then the sound pressure level at the outlet monitoring point was calculated by solving the Ffowcs Williams and Hawkings (FW-H) equation. The sound pressure level (SPL) spectrum at the monitoring point was obtained by fast Fourier transform (FFT), and the  
Figure 3: Air cavity grid of exhaust tailpipe: (**a**) prototype, (**b**) bionic exhaust pipe used for aeroacoustics, (**c**) prototype used for transmission loss, (**d**) bionic exhaust pipe used for transmission loss. Figure 2: Three-dimensional diagram of air cavity: (**a**) prototype and (**b**) bionic exhaust tailpipe. reference pressure was 2 \(\times\) 10\({}^{-5}\) Pa. The time step size was set to 0.0002 s, and the upper limit frequency was 2500 Hz to calculate the aeroacoustics. The simulation software LMS Virtual Lab Acoustics FEM was used to perform simulation analysis. The fluid was air, its density was 1.225 kg/m\({}^{3}\), and the velocity of the sound in the fluid was 340 m/s. The inlet boundary conditions were set as follows. The vibration velocity was 1 m/s, and the fluid velocity was 50 m/s. The full sound absorption property was set at the outlet to simulate no reflection boundary condition. The acoustic impedance of the fluid was \(\rho_{0}c=416.5\)kg/ (m\({}^{2}\cdot\)s), the calculation frequency was 10 Hz to 1000 Hz, and the calculation interval step was 20 Hz. The governing equations for LES are obtained by filtering Navier-Stokes equations in wavenumber space or physical space. The filter variable is defined as:  
\[\overline{\varphi}(x)=\int_{D}\varphi\left(x^{\prime}\right)G\left(x,x^{\prime }\right)dx^{\prime} \tag{1}\]  
where \(D\) is the flow area, \(\varphi\) is a transient variable, \(x^{\prime}\) is the actual flow area with spatial coordinates, and \(x\) is the spatial coordinates in the filtered large-scale space. \(G\) is the filter equation. The filter function to average physical quantities on the control volume is defined as:  
\[\overline{\varphi}(x)=\frac{1}{V}\int_{V}\varphi\left(x^{\prime}\right)dx^{ \prime},x^{\prime}\in v \tag{2}\]  
where \(V\) is the volume of the calculated unit. \[G\left(x,x^{\prime}\right)=\left\{ \begin{array}{l}1/V,x^{\prime}\in v\\ 0,x^{\prime}\
otin v\end{array} \right. \tag{3}\]  
The transient Navier-Stokes equation and continuous equation are processed by the filter function. \[\frac{\partial\left(\rho\right)}{\partial t}+\frac{\partial\left(\rho\overline {u}_{i}\right)}{\partial x_{i}}=0 \tag{4}\]  
\[\frac{\partial\left(\rho\overline{u}_{i}\right)}{\partial t}+\frac{\partial \left(\rho\overline{u}_{i}\overline{u}_{j}\right)}{\partial x_{j}}=\frac{ \partial\left(\sigma_{ij}\right)}{\partial x_{j}}-\frac{\partial\overline{p} }{\partial x_{i}}-\frac{\partial\tau_{ij}}{\partial x_{j}} \tag{5}\]  
where \(\rho\) is the air density, \(\overline{u}_{i}\) is the filter velocity component \(x_{i}\) in the Cartesian coordinate system, \(\overline{p}\) is the filter pressure, \(\sigma_{ij}\) is the stress tensor caused by molecular viscosity, and \(\tau_{ij}\) is the subgrid scale stress. \[\sigma_{ij}\equiv\left[\mu\left(\frac{\partial\overline{u}_{i}}{\partial x_ {j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\right)\right]-\frac{2}{ 3}\mu\frac{\partial\overline{u}_{i}}{\partial x_{j}}\delta_{ij} \tag{6}\]  
\[\tau_{ij}=\rho\overline{u_{i}u}_{j}-\rho\overline{u}_{i}\overline{u}_{j} \tag{7}\]  
In this research, the turbulent eddy viscosity is defined by the Wall-Adapting Local Eddy-Viscosity Model (WALE).