## 3 Governing Equations and Numerical Implementations
### Acoustic Analogy for Cavitation Noise Prediction  
Based on cavitation flow fields predicted using each of the above three viscous treatment approaches, the radiated cavitation noise from a hydrofoil is predicted by the FW-H equation [32]. In this study, so-called hybrid techniques to predict hydro-acoustic pressure waves are employed [33; 34; 35; 36]. The hybrid techniques consist of three steps; (1) flow field predictions by solving full 3D compressible RANS equations; (2) hydro-acoustic source modelling from the flow field results based on FW-H equation; and (3) linear wave propagation. Note that the present hybrid method does not consider the scattering and refraction of acoustic waves due to changes in sound velocity of media between source and acoustic regions. Therefore, the predicted hydro-acoustic fields are only used to investigate the relative contributions of the flow noise sources to the radiated acoustic field with emphasis on the role of cavitation flow characteristics due to viscous flux vectors.  
The wave equation, that accounts for aerodynamic noise sources, was formulated for the first time by [PERSON], who attempted to find the origin of jet noise. [PERSON] rearranged the Navier-Stokes equations into a wave equation with quadrupole source terms that were related to turbulent velocity fluctuations [37; 38]. [PERSON] [39] expanded [PERSON]'s acoustic analogy to include the effects of stationary solid bodies in flow and found the dipole source term that accounts for the fluid loadings on body surfaces. Finally, [PERSON] and [PERSON] considered the effects of arbitrarily moving solidbodies and showed that these effects could be modeled by using the monopole and dipole source terms, including the Doppler effect. The integral form of the FW-H equation is written as follows  
\[4\pi c^{2}H\rho^{\prime}\left(\overset{\rightarrow}{x},t\right)=\frac{\partial}{ \partial t}\int_{S}\frac{\rho_{0}\overset{\rightarrow}{v}\cdot\overset{ \rightarrow}{n}}{r|1-M_{r}|}dS\left(\overset{\rightarrow}{\eta}\right)- \frac{\partial}{\partial x_{i}}\int_{S}\frac{p_{ij}n_{j}}{r|1-M_{r}|}dS\left( \overset{\rightarrow}{\eta}\right)+\frac{\partial^{2}}{\partial x_{i} \partial x_{j}}\int_{V}\frac{T_{ij}}{r|1-M_{r}|}d^{3}\overset{\rightarrow}{\eta} \tag{18}\]  
where \(r\) is distance between receiver and source, \(M_{r}\) is the relative Mach number, and \(p_{ij}\) is a stress tensor defined as \(p_{ij}=p\delta_{ij}-\tau_{ij}\), where \(p\) is static pressure, \(\delta_{ij}\) is the Kronecker delta, and \(\tau_{ij}\) is a viscous stress tensor. [PERSON]'s stress tensor, \(T_{ij}\), is defined as \(T_{ij}=\rho u_{i}u_{j}+\left(p-c_{0}^{2}p\right)\delta_{ij}-\tau_{ij}\).  
By following the procedure from a previous study ([PERSON] et al. (2016)), the FW-H equation can be rewritten in a form appropriate to account for flow noise, due to cavitation flow, over an acoustically compact stationary body, as follows  
\[p^{\prime}(x,t)=\frac{\rho_{0}}{4\pi}\frac{\partial}{\partial t}\left[\frac{1 }{r}\int_{V}\frac{\partial\alpha_{v}}{\partial t}\,dV\right]-\frac{\partial} {\partial x_{i}}\int_{S}\frac{p_{ij}n_{j}}{4\pi r}dS \tag{19}\]  
where \(\alpha_{v}\) is the vapor volume fraction.  
The first term on the right-hand side of Equation (19) is the acoustic pressure due to the monopole source, caused by the rate of cavitation volume changes, and the second term is the dipole source of hydro-dynamic forces on the body surface.