# Investigation on Characterization of Typical Characteristic in Compressor Based on Flat Plate Model
## 2 Flow-Induced Noise Theory  
Simulation of the flow state is conducted in the present investigation. Initially, the steady flow field calculation is carried out to achieve the initialization of the flow state and then is used as the input condition for the transient flow field calculation. Renormalization-group (RNG) \(k-\varepsilon\) model is taken for the steady-state calculation, the small scale motion in the flow is no longer calculated separately but is represented in the large scale motion and viscous motion. Therefore, the governing equations are not considered with the small scale motion. \(k\) equation and \(\varepsilon\) equation can be expressed as  
\[\left\{ \begin{array}{l}\frac{\partial(\rho k)}{\partial t}+\frac{\partial( \rho ku_{i})}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\Big{[}\frac{ \alpha_{k}H_{eff}\partial k}{\partial x_{j}}\Big{]}+G_{k}+\rho\varepsilon\\ \frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho cu_{i})}{ \partial x_{i}}=\frac{\partial}{\partial x_{j}}\Big{[}\frac{\alpha_{k}H_{eff }\partial\varepsilon}{\alpha_{j}}\Big{]}+\frac{C_{k}^{\dagger}cG_{k}}{k}- \frac{C_{2}\rho\varepsilon^{2}}{k}\end{array} \right. \tag{1}\]  
where \(\rho\) is the flow density, \(u_{i}\) and \(u_{j}\) are the flow velocity. \(G_{k}\) is the turbulent kinetic energy generated by the laminar velocity gradient. For the transient state calculation, the large eddy simulation (LES) turbulence model is selected in the paper [31; 32]. After processing through the mathematical filter function, the governing equation LES is described by  
\[\left\{ \begin{array}{l}\frac{\partial}{\partial t}(\rho\overline{u_{i}})+ \frac{\partial}{\partial x_{j}}\big{(}\rho\overline{u_{i}u_{j}}\big{)}=-\frac {\partial p}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Big{(}\mu\frac{ \partial\overline{u_{i}}}{\partial x_{j}}\Big{)}-\frac{\partial\tau_{ij}}{ \partial x_{j}}\\ \frac{\partial\rho}{\partial t}+\frac{\partial u}{\partial x_{i}}(\rho \overline{u_{i}})=0\end{array} \right. \tag{2}\]  
where the parameter with the overbar '\(\leftarrow\)' is the field variable after filtering. \[\tau_{ij}=\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}-\rho\overline{u_{i}^{ \prime}u_{j}^{\prime}} \tag{3}\]  
The above formula is the subgrid-scale stress (SGS), which is associated with the small scale motion in the motion equation. The variable value is the instantaneous value after filtering, which is different from the time-averaged value of Reynolds-averaged Navier-Stokes (RANS). The stress term in SGS is unknown which needs to be derived from other physical quantities. The basic model of SGS is proposed by [PERSON] [33]:  
\[\tau_{ij}-\frac{1}{3}\tau_{ik}\delta_{ij}=-2\mu_{t}\overline{S_{ij}} \tag{4}\]  
where \(\mu_{t}\) is the turbulent viscosity of the sublattice scale model, \(\mu_{t}=(C_{s}\Delta)^{2}|\overline{S}|\). \[\left\{ \begin{array}{l}\overline{S_{ij}}=\frac{1}{2}\Big{(}\frac{\partial u _{i}}{\partial x}+\frac{\partial u_{j}}{\partial x_{i}}\Big{)}\\ |\overline{S}|=\sqrt{2S_{ij}S_{ij}}\\ \Delta=\big{(}\Delta_{x}\Delta_{y}\Delta_{z}\big{)}^{1/3}\\ C_{s}=\frac{1}{\pi}\big{(}\frac{2}{3}C_{x}\big{)}^{3/4}\end{array} \right. \tag{5}\]  
where \(C_{x}\) is the Smagorinsdky constant. The flow characteristics of the vortex group in the cavity of the structure are captured by computational fluid dynamics (CFD) which is based on the above equations. In the paper, the sound source information is extracted from the flow field information by the Lighthill acoustic analogy method. The mass conservation equation and the momentum conservation equation can be expressed as:  
\[\left\{ \begin{array}{l}\frac{\partial\rho}{\partial t}+\frac{\partial\rho v_{ i}}{\partial x_{i}}=0\\ \frac{\partial\rho v_{i}}{\partial t}+\frac{\partial\rho v_{i}v_{j}}{\partial x _{j}}=-\frac{\partial p_{ij}}{\partial x_{i}}\end{array} \right. \tag{6}\]  
Additionally,  
\[p_{ij}=p\delta_{ij}-\tau_{ij} \tag{7}\]  
\(\rho\), \(p\), and \(v\) are the density, pressure, and velocity, respectively, of the flow in the presence of acoustic disturbance.