# The Characteristics and Mechanisms of High-Intensity Sound in a High-Speed Multistage Compressor
## 2 Numerical Calculation Theory  
### Flow-Induced Sound Theory  
During the calculation, the renormalization group (RNG) \(k-\varepsilon\) model is selected to calculate the steady-state flow field. The LES turbulence model is taken in the present investigation for transient flow field calculations [31]. After addressing the flow field, the Lighthill acoustic analogy method is taken to derive the sound source information from the flow field information [32]. The mass conservation and momentum conservation equations are as follows:  
\[\left\{\begin{array}{l}\frac{\partial\rho}{\partial t}+\frac{\partial\rho v _{i}}{\partial x_{i}}=0\\ \frac{\partial\rho v_{i}}{\partial t}+\frac{\partial\rho v_{i}v_{j}}{\partial x _{j}}=-\frac{\partial v_{ij}}{\partial x_{i}}\end{array}\right. \tag{1}\]In the above equation, the term \(p_{ij}\) can be expressed as:  
\[p_{ij}=p\delta_{ij}-\tau_{ij} \tag{2}\]  
where \(\rho\), \(p\) and \(v\) are the density, pressure and velocity in the presence of acoustic disturbances in the flow. \(\delta_{ij}\) is the Kronecker notation (\(\delta_{ij}=1\), \(\delta_{ij}=0\); \(i\
eq j\)).  
Taking the time and space partial derivatives of the formula in Equation (1), the following can be obtained:  
\[\frac{\partial^{2}\rho}{\partial t^{2}}=\frac{\partial}{\partial x_{i}}\left( \frac{\partial\rho v_{i}v_{j}}{\partial x_{j}}+\delta_{ij}\frac{\partial p}{ \partial x_{i}}+\frac{\partial\tau_{ij}}{\partial x_{j}}\right) \tag{3}\]  
Subtracting \(c_{0}{}^{2}\frac{\partial^{2}\rho}{\partial x_{i}{}^{2}}\) from both sides of Equation (3) simultaneously:  
\[\frac{\partial^{2}\rho}{\partial t^{2}}-c_{0}{}^{2}\frac{\partial^{2}\rho}{ \partial x_{i}{}^{2}}=-\frac{\partial^{2}T_{ij}}{\partial x_{i}\partial x_{j}} \tag{4}\]  
The parameters of Equation (4) satisfy the following relationship:  
\[\left\{\begin{array}{l}T_{ij}=\rho v_{i}v_{j}-(p-c_{0}\rho)\delta_{ij}+\tau _{ij}\\ p=p_{0}+p_{a}\\ \rho=\rho_{0}+\rho_{a}\end{array}\right. \tag{5}\]  
where \(p_{0}\) and \(\rho_{0}\) are the airflow pressure and density when there are no sound, \(p_{a}\) and \(\rho_{a}\) are the pressure and density changes arising from the pulsation of sound pressure, and \(c_{0}\) is the sound velocity separate from the sound source and the mean flow area. \(T_{ij}\) is the Lighthill stress tensor.  
Due to \(\frac{\partial^{2}\rho_{0}}{\partial t^{2}}-c_{0}{}^{2}\frac{\partial^{2}\rho _{a}}{\partial x_{i}x_{i}}=0\), the simultaneous Equation (4) is derived as:  
\[\frac{\partial^{2}\rho_{a}}{\partial t^{2}}-c_{0}{}^{2}\frac{\partial^{2}\rho _{a}}{\partial x_{i}\partial x_{i}}=-\frac{\partial^{2}T_{ij}}{\partial x_{i} \partial x_{j}} \tag{6}\]  
At low Mach numbers and small amplitudes, regardless of the entropy source terms and viscous stress conditions, the following relation can be obtained:  
\[\rho v_{i}v_{j}=(\rho_{0}+\rho_{a})v_{i}v_{j}\approx\rho_{0}v_{i}v_{j}=T_{ij} \tag{7}\]  
Therefore, the Lighthill acoustic analogy equation is derived as follows:  
\[\left\{\begin{array}{l}\frac{\partial^{2}\rho_{a}}{\partial t^{2}}-c_{0}{}^ {2}\frac{\partial^{2}\rho_{a}}{\partial x_{i}\partial x_{j}}=-\frac{\partial^ {2}T_{ij}}{\partial x_{i}\partial x_{j}}\\ \frac{\partial^{2}p}{\partial x_{i}\partial x_{i}}=\frac{1}{c_{0}{}^{2}}\frac{ \partial^{2}p}{\partial t^{2}}\end{array}\right. \tag{8}\]  
This method considers the flow field separately from the sound field. In the first equation of Equation (8), the right term can be regarded as the source term which can be calculated from the flow field, and the left is a typical sound wave equation.