# Research on Characteristics of Flow Noise and Flow-Induced Noise
## 2 Computational Theory  
### Flow Noise Calculation Theory  
The study of flow noise started with [PERSON]'s acoustic analogy theory, which combines acoustic field and flow field calculations to solve flow noise by solving flow field information [16]. [PERSON]'s basic equation is as follows:  
\[\frac{\partial^{2}\rho^{\prime}}{\partial t^{2}}-c_{0}^{2}\
abla^{2}\rho^{ \prime}=\frac{\partial^{2}T_{ij}}{\partial x_{i}\partial x_{j}} \tag{1}\]  
where \(c_{0}\) is the speed of sound under isentropic conditions; \(\rho^{\prime}=\rho-\rho_{0}\), where \(\rho\) and \(\rho^{\prime}\) are the densities of the fluid under disturbed and undisturbed conditions, respectively; \(T_{ij}\) is the [PERSON]'s stress tensor. The boundary element method is capable of accurately computing and solving flow noise for complex structures at low frequencies. By utilizing Helmholtz equation in conjunction with the acoustic control equation, the differential equations within the solution domain are transformed into integral equations over the boundary. These integral equations are then discretized into simpler weighted forms using numerical methods [17]. Let \(\hat{p}^{\prime}e^{i\omega t}=c^{2}\hat{\rho}e^{i\omega t}=c_{0}^{2}\hat{p}^{ \prime}\), \(\hat{q}=\frac{\hat{\sigma}^{T}\hat{\tau}_{ij}}{\partial x_{i}\partial x_{j}}\), \(T_{ij}=\hat{T}_{ij}e^{i\omega t}\), where \(k=\omega/c_{0}\) is the wave number, the symbol above \(\hat{p}^{\prime},\hat{\theta}^{\prime}\) and \(\hat{T}_{ij}\) indicates that they are frequency domain variables. Taking the Fourier transform of Equation (1), we arrive at the following expression:  
\[\
abla^{2}p^{\prime}+k^{2}p^{\prime}=q \tag{2}\]  
For the sake of simplicity, the symbols denoting frequency domain variables have been omitted. In the subsequent text, unless otherwise specified, all variables represent frequency domain quantities. Reference [18] applies a specific treatment to Equation (2), indicating that the sound pressure at the receiving point in the acoustic field needs to satisfy the following formula:  
\[Cp^{\prime}=\int_{V}T_{ij}\frac{\hat{\sigma}^{2}G}{\partial y_{i}\partial y_{ j}}\mathrm{d}V-\int_{\Omega_{a}}p^{\prime}\frac{\partial G}{\partial n} \mathrm{d}\Omega_{a} \tag{3}\]  
where \(V\) is the volume domain to be solved; \(\Omega_{a}\) is the solid wall surface; \(C\) is the surface shape function, taking 0.5 when the receiving point is on the solid surface and 1. when the receiving point is inside the solution domain; \(G\) is the Green's function satisfying \(G(\overrightarrow{r},\overrightarrow{r}_{a})=\frac{e^{-|R|\overrightarrow{r}- \overrightarrow{r}_{a}|}}{4\pi|\overrightarrow{r}-\overrightarrow{r}_{a}|}\), where \(r_{a}\) is the sound source point. In Equation (2), the flow noise \(p^{\prime}\) can be decomposed into the sum of hydrodynamic fluctuating pressure \(p_{h}\) and acoustic fluctuating pressure \(p_{a}\), i.e.,  
\[p^{\prime}=p_{h}+p_{a} \tag{4}\]  
where \(p_{h}\) can be obtained by calculating the flow field, but \(p_{a}\) is unknown. To solve for \(p_{a}\), the integral domain needs to be decomposed. The volume domain is decomposed into \(V_{1}\) and \(V_{2}\), with corresponding boundaries of \(\Omega_{1}\) and \(\Omega_{2}\), respectively. \(V_{1}\) is the volume close to the receiving point \(r\), which is only affected by \(p_{a}\); \(\Omega_{2}\) is the remaining volume of the integral domain, where the receiving point \(r\) is affected by both \(p_{a}\) and \(p_{h}\). Substituting Equation (4) into (3) and using the approximate formula of Lighthill stress tensor \(T_{ij}\approx\rho_{0}u_{i}u_{j}\), we can obtain:  
\[Cp_{a}=-\int_{\Omega}p_{a}\frac{\partial G}{\partial n}\mathrm{d}\Omega+\int _{V_{2}}\rho_{0}u_{i}u_{j}\frac{\hat{\sigma}^{2}G}{\partial y_{i}\partial y_{ j}}\mathrm{d}V_{1}-\int_{\Omega_{2}}p_{h}\frac{\partial G}{\partial n} \mathrm{d}\Omega_{2} \tag{5}\]  
Before solving using the boundary element method, discretization is first performed.