## 2 Governing Equations and Numerical Methods
### Acoustic Power Due to Internal Flow Noise  
As described in the Introduction, to find the precise mechanism of AIV and to devise its effective measure, the first-principle-based method is essential. [PERSON] and [PERSON] [7] used a fluid-structure one-way coupling method for fluid-structure interaction based on ACTRAN software which uses the finite element method and suggested a methodology for AIV risk assessment. However, the method is still based on an empirical formula of [PERSON] for the prediction of acoustic power. In addition to this study, most of the previous studies for the design criteria to prevent the fatigue failure caused by the AIV depend on the simple empirical formula by [PERSON] and [PERSON]. In this section, the formula for acoustic power due to internal flow noise is presented using the wavenumber-frequency transform of pressure field in time-space domains and the duct acoustic theory. As a starting point, the pressure fluctuation due to the flow on the pipe wall needs to be separated into the incompressible and the compressible pressure fluctuations [13]. The incompressible pressure fluctuation is not associated with the density perturbation of the fluid. It is also called the hydrodynamic pressure fluctuation or the pseudo sound and propagates at the same speed as the convection eddy velocity in the fluid. Compressible pressure fluctuation refers to acoustic waves due to perturbations of density and velocity in a fluid and propagates at the speed of sound. A wavenumber-frequency analysis can be used to separate the incompressible and compressible components from the total pressure fluctuation components on the wall. Since the target problem of this study is a circular pipe, the Fourier transform in the cylinder coordinate system is more suitable. From the previous study, the wavenumber-frequency analysis is performed in the form [8],  
\[S(m,\ p\Delta k_{z},o\Delta f)\ =\ \frac{1}{N_{t}N_{z}N_{\theta}}\sum_{n\ =\ 0}^{\infty}\frac{1}{J_{m}(k_{r_{0},m,n}\ r_{0})}\sum_{k\ =\ 1}^{N_{t}-1}\sum_{l\ =\ 0}^{N_{t}-1}\sum_{j\ =\ 0}^{N_{t}-1}w_{klj}\ p_{klj}e^{-2\pi i(\frac{k_{r}}{N_{t}}+\frac{m}{N_{ \theta}}-\frac{ip}{N_{\pi}})} \tag{9}\]  
where \(S\) is the spectral density, \(m\) is the circumferential mode, \(n\) is the radial mode, \(k_{z}\) is the axial direction wavenumber, and \(J_{m}\) is the Bessel function of \(m\)th order. \(N\) means the size of the discretized signal over time and space. The Hanning function is used as a window. The axial wavenumber, \(k_{z}\), from duct acoustics is defined as:  
\[k_{z,m,n}^{\pm}\ =\ \frac{\mp\ Mk_{0}\ +\ \left[k_{0}^{2}-\left(1-M^{2}\right) k_{r,m,n}^{2}\right]^{1/2}}{1-M^{2}} \tag{10}\]  
\[k_{0}^{2}\ -\ \left(1\ -\ M^{2}\right)k_{r,m,n}^{2}\ >\ 0 \tag{11}\]  
The Fourier transform of the discretized pressure fluctuation signal in time and space can be used to obtain a wavenumber-frequency diagram from Equation (9). The acoustic power generated in the pipe can be obtained by defining the instantaneous sound intensity corresponding to the acoustic power per unit area and integrating it over the area. Since the empirical approach uses acoustic power as a criterion to determine the possible occurrence of AIV, the formula for the acoustic power is derived and presented combining the wavenumber-frequency spectrum with the duct acoustic theory. The sound pressure and particle velocity in the pipe can be deduced from the duct acoustic theory [14],  
\[p(r,\ \theta,\ z,\ t)\ =\ \sum_{m\ =\ 0}^{\infty}\sum_{n\ =\ 0}^{\infty}J_{m}(k_{r,m,n}r)e^{im \theta}e^{ivt}\times\left(C_{+,m,n}e^{-jk_{z,m,n}^{+}\ z}+C_{-m,n}e^{+jk_{z,m,n }^{z}}\right) \tag{12}\]  
\[u_{z}(r,\ \theta,\ z,\ t)\ =\ \sum_{m\ =\ 0}^{\infty}\sum_{n\ =\ 0}^{\infty}\frac{k_{z,m,n}^{+}}{k_{0}\rho_{0}c_{0}}J_{m}(k_{r,m,n}r)cos(m \theta)C_{+,m,n}e^{i(\alpha t-k_{z,m,n}^{+}\ z)} \tag{13}\]  
The acoustic intensity can be expressed as the product of the compressible pressure fluctuation obtained previously and the particle velocity. Finally, it can be expressed as  
\[