# Study on Noise Model of an Automotive Axial Fan Based on Aerodynamic Load Force
## 3 Noise Model of Automotive Axial Fan
### Establishment of Fan Mathematical Model Noise  
The FW-H equation is the basic equation of aeroacoustics. Assuming that the control surface \(f(x_{i},\ t)=0\) is the fan blade surface, the most commonly used form of the FW-H equation can be obtained:  
\[\left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{ \partial x_{i}^{2}}\right)p(x_{i},\ t)=\frac{\partial[\rho_{0}v_{n}\delta(f)]} {\partial t}-\frac{\partial[F_{j}\delta(f)]}{\partial x_{i}}+\frac{\partial^{ 2}[T_{ij}H(f)]}{\partial x_{i}\partial x_{j}} \tag{31}\]  
In the formula, \(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{ \partial x_{i}^{2}}\) is the operator for volatility, \(p(x_{i},\ t)\) represents the sound pressure value of the observation point at time \(t\), \(v_{n}=-\frac{\partial f}{\partial t}\) represents the outside normal vector of the control surface velocity, \(F_{i}=-P_{ij}\cdot\eta_{i}\) is the blade surface load component, and the far right term represents the thickness sound source, load sound source, and quadrupole sound source, respectively. \(T_{ij}=\rho u_{i}u_{j}-P_{ij}-c^{2}\rho^{\prime}\delta_{ij}\) is the Lighthill stress tensor. \(\rho\) and \(P_{ij}\) are the density and stress tensors, respectively. \(\delta_{ij}\) is Kronecker symbol; subscript 0 represents undisturbed quantity; the apostrophe denotes perturbations; \(n\) represents the projection of control surface normal; \(H(f)\) is the Heaviside function; and \(\delta(f)\) represents Dirac and satisfies  
\[H(f)=\left\{ \begin{array}{l}1\frac{f(x_{i},\ t)>0}{f(x_{i},\ t)<0}\,,\ \delta(f)=x= \frac{\partial H(f)}{\partial f}\end{array} \right. \tag{32}\]  
The automotive axial fan is at low speed and subsonic flow, so the quadrupole noise can be ignored. Assuming that the volume density is constant, the non-equilibrium mass flow into the fluid is equivalent to a monopole, which can be ignored. In this case, the FW-H equation can be expressed as  
\[\left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{ \partial x_{i}^{2}}\right)p(x_{i},\ t)=\frac{\partial[\rho_{0}v_{n}\delta(f)]} {\partial t}-\frac{\partial[F_{j}\delta(f)]}{\partial x_{i}} \tag{33}\]  
It is assumed that the fan blades are in constant motion. In fan blade noise calculated by the application of delay time, the integrand function of each sound source on the blade surface \(f=0\) is adopted using the appropriate value corresponding to the respective delay time \(\tau\), namely the acoustic pressure \(p(x,\ t)\), the sound blade surface microelements in their respective corresponding delay time \(\tau\) emissions, and the sum of the acoustic signals of the same receiving points in \(X\). According to FW-H Equation (33), the time-domain calculation formula of noise is \[p(x,\ t)=\frac{1}{4\pi}\int_{f=0}\left[\frac{\rho_{0}V_{n}\left(r\dot{M}_{i}\dot{r} _{i}+c_{0}M_{r}-c_{0}M^{2}\right)}{r^{2}(1-M_{r})^{3}}\right]_{ret}\mathrm{d}s +\frac{1}{4\pi c_{0}}\ \int_{f=0}\left[\frac{\dot{t}_{i}\dot{r}_{i}}{r(1-M_{r})^{2}} \right]_{ret}\mathrm{d}s+\frac{1}{4\pi}\ \int_{f=0}\left[\frac{\dot{t}_{i}\dot{r}_{i}}{r^{2}(1-M_{r})^{2}} \right]_{ret}\mathrm{d}s+ \tag{34}\] \[\frac{1}{4\pi c_{0}}\ \int_{f=0}\left[\frac{\dot{t}_{i}\left(r\dot{M}_{i} \dot{r}_{i}+c_{0}M_{r}-c_{0}M^{2}\right)}{r^{2}(1-M_{r})^{3}}\right]_{ret} \mathrm{d}s\]  
In the formula, the first term is the thickness noise, the second, third, and fourth terms are the load noise, and \"_ret_\" stands for the delay time. \(\rho_{0}\) is the air density, and \(c_{0}\) is the sound velocity. \(V_{i}\) is the velocity component of the particle velocity fixed on the surface along the \(x_{i}\) axis and the subscript \(i\) represents the component along the \(x_{i}\) axis. \(V_{n}=V_{i}n_{i}\) is the normal velocity of the moving elements along the physical plane and \(n_{i}\) is the normal vector of the elements. \(\dot{V}_{n}\) is the derivative of the normal velocity of the moving elements along the physical plane with time.