## 2 Background Information
### Generation Mechanism of Hydrodynamic Noise
#### 2.1.2 Wake Noise  
When a fluid flows past an object, the flow separates near the object's surface, creating a region of vorticity known as the \"wake region\". As demonstrated in Figure 2, in the wake region, the flow becomes unstable, leading to the formation of vortices that shed periodically. The periodicity and intensity of vortex shedding are contingent on the flow velocity, the object's inherent dimensions and geometry, and the fluid's viscosity. Each vortex shedding event converts disturbances in the flow into sound waves, thereby producing wake noise. The predominant component of wake noise is low-frequency noise, which is attributable to the generally low vortex shedding frequency. Furthermore, vortex shedding not only alters the structure of the flow field but also generates sound waves that propagate through the air and radiate into the surrounding space. In a study by [PERSON] [30], based on viscous multiphase flow theory, the Navier-Stokes equation, a turbulence model, and a cavitation model were applied to numerically predict the vapor volume fraction and pressure in a propeller cavitation wake. The investigation further delved into the correlation between the characteristics of the cavitation wake and the ensuing cavitation noise. The vortex shedding frequency \(f\) is associated with the flow velocity \(U\) and the object's characteristic size \(d\). The periodicity of vortex shedding is indicative of the instability of the flow, and its frequency is typically inversely proportional to the size of the object and  
Figure 2: Numerical and experimental results of vortex shedding [31]: (**a**) the velocity distribution; (**b**) the flow state at the outlet. proportional to the flow velocity. The frequency of vortex shedding can be described by the Strouhal number, which is defined as follows:  
\[St\ =\ \frac{fd}{U} \tag{3}\]  
where  
\(f\): Vortex shedding frequency,  
\(d\): The characteristic size of an object, usually the diameter or length of the object,  
\(U\): Flow rate. As demonstrated by this Formula (3), the shedding frequency of the vortex is directly and inversely proportional to the flow velocity and the characteristic size of the object. That is to say, the greater the flow velocity, the higher the shedding frequency of the vortex. Conversely, the shedding frequency of the vortex is lower when the size of the object is larger. In the context of underwater vehicles, particularly in the context of tail design optimization, effective control of the phenomenon of vortex shedding can be achieved. This reduction in noise is a key benefit of these designs. However, the eddy current signal is often superimposed by other noises, such as factory noise and measurement noise. To effectively extract the vortex signal, [PERSON] [32] proposed the use of a recursive filter to estimate the vortex shedding frequency. The implementation of a Kalman filter served to ensure the smoothening of the vortex shedding signal, thereby facilitating the precise calculation of the flow rate within the turbulent flow range. The shedding of vortices has been shown to alter the structure of the flow and to initiate sound waves through the momentum source. To quantitatively describe the intensity of the wake noise, the acoustic power level of the noise can be derived by combining the intensity of the vortex with the shedding frequency. The strength of the vortex is generally proportional to the square of the flow velocity and is closely related to the characteristic size of the object, the viscosity of the fluid, and the turbulence intensity of the flow. The sound power level of the wake noise can be estimated by the following Equation (4):  
\[L_{w}\ =\ 10\log(\frac{U^{3}}{vd}) \tag{4}\]  
where  
\(L_{w}\): Sound power level of wake noise in dB,  
\(U\): Flow rate,  
\(d\): The characteristic size of the object (usually the diameter of the object),  
\(\
u\): Dynamic viscosity of the fluid. As demonstrated by the aforementioned Equation (4), the impact of flow velocity \(U\) on wake noise is of paramount importance.