## 2 Background Information
### Generation Mechanism of Hydrodynamic Noise
#### 2.1.1 Turbulent Boundary Layer Noise  
Turbulent boundary layer noise is caused by pressure pulsations in the turbulent boundary layer, formed by the fluid on the surface of the navigational body. As illustrated in Figure 1, the motion of fluid particles in the turbulent boundary layer under high-Reynolds-number conditions is highly random, forming a large number of vortices [23]. The interaction of these vortices, in conjunction with the friction experienced at the boundary surface, gives rise to pressure fluctuations, which, in turn, generate noise. It has been observed that when the flow velocity is high or the fluid viscosity is low, the turbulence within the boundary layer intensifies, resulting in an elevated noise level. The intensity of turbulent boundary layer noise is influenced by various factors, including the fluid velocity, temperature, viscosity, the roughness of the boundary surface, and the interaction between the fluid and the solid interface. Specifically, the smoothness of the boundary surface has been shown to have a substantial impact on turbulence characteristics, consequently altering the intensity and frequency characteristics of the noise [24]. [PERSON]'s acoustic analogy is a foundational framework within the theoretical study of turbulent noise. He proposed that, within the context of turbulence, sound waves are initiated by momentum sources, specifically velocity fluctuations and disturbances. Figure 1: A high-Reynolds-number turbulent boundary layer moving from left to right [25]. These momentum sources propagate into space through nonlinear interactions, eventually forming sound waves. The generation process of turbulent noise can thus be derived through the following steps:  
1. The introduction of momentum sources is an essential component of the study. In the context of turbulence, local flow irregularities, such as velocity fluctuations, serve as the underlying momentum sources. [PERSON] drew parallels between this process and the generation of a sound source, emphasizing that the strength of the former is frequently associated with the energy of the turbulence and proportional to the square of the flow velocity. 2. The relationship between the momentum source and the sound pressure is as follows: In the context of turbulence, momentum sources excite sound waves through fluid disturbances. The relationship between sound pressure and momentum sources is governed by [PERSON]'s acoustic analogy, which is often expressed in the form of the Lighthill Equation (1). \[\frac{\partial^{2}p}{\partial t^{2}}-c^{2}\
abla^{2}p\ =\ \frac{\partial}{ \partial x_{i}}(T_{ij}\frac{\partial}{\partial x_{j}})\] (1) where \(p\): Sound pressure, \(c\): Speed of sound, \(\
abla^{2}\): Laplacian, representing the second derivative in space, \(T_{ij}\): Stress tensor of Light hill, representing the source of turbulence. In the context of turbulence, the stress tensor \(T_{ij}\) is a critical component, as it represents the momentum source of the turbulence. The solution to this equation provides insight into the intensity of the noise generated by the turbulence. 3. The following is a derivation of the simplified formula: In practical calculations, the turbulent noise intensity is usually related to the turbulence characteristics (e.g., boundary layer thickness, flow velocity) and the viscosity of the fluid. According to the principles of fluid dynamics and turbulence theory, the turbulent boundary layer noise intensity can be expressed by the following simplified Formula (2): \[L_{w}\ =\ 10\text{log}\bigg{(}\frac{d}{L}\bigg{)}+10\text{log}\bigg{(}\frac{U}{v} \bigg{)}+C\] (2) where \(d\): The characteristic length of the object, \(L\): Thickness of the boundary layer, \(U\): Flow rate, \(\
u\): Dynamic viscosity, \(C\): Constant, usually around 30. The Formula (2) for \(L_{w}\) is indicative of the impact of turbulent boundary layer dimensions on noise.