## 3 Study Methodologies
### Theoretical Study of Hydrodynamic Noise  
Theoretical studies analyze the flow-induced noise through the governing equation level. The flow and sound fields have the same governing equation and the parameters of flow and sound fields could be obtained simultaneously by solving the compressible Navier-Stokes (N-S) equation directly in theory. However, it is difficult realize due to the huge difference in scale and magnitude between flow and sound fields [37]. Therefore, relevant sound theory is used. Table 2 summarizes the comparison of results between Lighthill acoustic analogy theory, vortex sound theory and acoustic/viscous splitting theory.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sound Theory** & \begin{tabular}{c} **Lighthill Acoustic Analogy** \\ **Theory**[38; 39; 40] \\ \end{tabular} & **Vortex Sound Theory**[41; 42; 43; 44; 45; 46] & **Acoustic/Viscous Splitting Theory**[47; 48] \\ \hline \multirow{4}{*}{\begin{tabular}{c} Governing \\ equation \\ \end{tabular} } & \begin{tabular}{c} Inhomogeneous sound wave \\ equation with sound source. \\ The left and right side items of \\ the sound wave equation are \\ wave operator and sound \\ source, respectively. \\ \end{tabular} & \begin{tabular}{c} Same as the form of the \\ governing equation of \\ \"Lighthill acoustic \\ analogy theory.\" \\ \end{tabular} & \begin{tabular}{c} Compressive perturbation \\ equation of sound field is \\ obtained by subtracting the \\ incompressible N-S equation \\ of flow field from compressible \\ N-S equation based on \\ variable separation method. \\ \end{tabular} \\ \hline \multirow{4}{*}{\begin{tabular}{c} Sound source \\ \end{tabular} } & \begin{tabular}{c} Monopole, dipole and \\ quadrupole sources. \\ \end{tabular} & \begin{tabular}{c} The stretching of the vortex \\ and the non-uniformity of \\ fluid kinetic energy. \\ \end{tabular} & / \\ \hline \multirow{4}{*}{\begin{tabular}{c} Advantages \\ \end{tabular} } & \begin{tabular}{c} It is suitable for the frequency \\ response analysis and the \\ distribution characteristics \\ analysis of sound field in \\ frequency domain and the \\ improved calculation process \\ has been formed. \\ \end{tabular} & \begin{tabular}{c} The time-domain parameters \\ of sound field could be \\ obtained directly. It is suitable \\ for the interpretation of the \\ generation and distribution of \\ flow-induced noise. \\ \end{tabular} & Same as the advantages of \\ \hline \multirow{4}{*}{\begin{tabular}{c} Shortages \\ \end{tabular} } & \begin{tabular}{c} Fluid viscosity is ignored. It \\ could not explain the \\ generation and distribution \\ mechanism of \\ flow-induced noise. \\ \end{tabular} & Fluid viscosity is ignored. The \\ \cline{1-1}  & \
\begin{tabular}{c} Fluid viscosity is ignored. The \\ solution process is relatively \\ complicated and the improved \\ calculation process has not \\ been formed. \\ \end{tabular} &
\begin{tabular}{c} The solution process is more \\ complicated than that of \\ “Vortex sound equation” and \\ the improved calculation process has not \\ process has not been formed. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison between Lighthill acoustic analogy theory, Vortex sound theory and Acoustic/Viscous splitting theory.