## 2 Background Information
### Numerical Simulation of Hydrodynamic Noise
#### 2.2.3 Reynolds-Averaged Navier-Stokes Equation (RANS)  
The Reynolds-averaged Navier-Stokes equation (RANS) is combined with a turbulence model to estimate the mean flow field and the turbulence contribution to the noise. Given that only the mean flow field and the associated turbulence characteristics need to be solved, this is currently the most commonly used numerical simulation method in engineering applications. Its computational efficiency makes it particularly suitable for fast estimation of flow noise in scenarios that do not require excessively high accuracy [65]. RANS is suitable for flow noise prediction in industrial and engineering scenarios, and it is particularly advantageous in the fast assessment of large-scale, domain-wide noise.  
The RANS method operates under the assumption that the flow is statistically homogeneous. It calculates a Reynolds-averaged equation by time-averaging the Navier-Stokes Equation (10):  
\[\frac{\partial\overline{u}}{\partial_{t}}+\overline{u}\
abla\overline{u}\ =\ - \frac{1}{\rho}\
abla\overline{p}+v\
abla^{2}\overline{u}+f+\
abla\tau_{R} \tag{10}\]  
where  
\(\overline{u}\): Average speed,  
\(\overline{p}\): Average pressure,  
\(\tau_{R}\): Reynolds stress.  
The Reynolds stress \(\tau_{R}\) is calculated using a turbulence model. A typical Reynolds stress model (11) is the Boussinesq assumption:  
\[\tau_{R}\ =\ -\rho\overline{u^{\prime}_{i}u^{\prime}}_{j}\ =\ \mu_{t}(\frac{ \partial\overline{u_{i}}}{\partial xj}+\frac{\partial\overline{u_{j}}}{ \partial x_{i}}) \tag{11}\]  
where \(\mu_{t}\) is the turbulent viscosity, usually provided by a turbulence model.  
The primary disadvantage of the RANS (Reynolds-averaged Navier-Stokes) model is its inability to capture transient variations in turbulence through its averaging computational process. This results in an inaccurate description of small-scale turbulence structure and noise sources. The applicability of RANS is limited in applications requiring high-precision noise prediction, prompting researchers to utilize RANS for preliminary estimation or in conjunction with other high-precision methods to enhance overall computational outcomes. In recent years, the hybrid LES-RANS method has been gradually applied in large-scale engineering projects, which can compensate for the insufficient capture of turbulent structural details by RANS to a certain extent. [PERSON] [66] compared the RANS and the hybrid RANS-LES methods with experimental data in a diffuser problem and noted that, although the pure RANS model, due to its high computational efficiency, remains the common choice for industrial applications, the hybrid RANS-LES method provides more accurate results in certain complex flow situations [67], especially when flow phenomena such as large-scale eddies, turbulent separation, etc., need to be considered. In order to enhance the precision and reliability of hybrid LES-RANS simulations, [PERSON] [68] developed an adaptive method to improve the vortex viscosity prediction in a hybrid large eddy simulation (LES)-RANS model by leveraging upstream LES results. [PERSON] [69] conducted a comparative test for three different operating conditions using a hybrid LES-RANS model with the traditional LES method. The study demonstrated that the results of the hybrid LES-RANS model were in good agreement with the experimental data. The hybrid LES-RANS model exhibited a key advantage over the traditional LES method, namely a reduction in computation time and mesh size.