# Research on a Strategy for Prediction Methods of Submarine Self-Propulsion Hydrodynamic Performance
## 2 Numerical Simulation Basics
### Turbulence Model  
Due to the highly complex nature of turbulent motion, it is extremely difficult to theoretically explore its mechanisms, and it is even more challenging to conduct comprehensive and accurate numerical simulations of all the flow details in turbulent motion. In computational fluid dynamics, attempts are generally made to approximate or simplify turbulent flows based on the primary flow characteristics of interest. Different turbulence models are selected for different flow problems to facilitate calculations and analyses [25]. The turbulence models commonly used in engineering today include the following types: simplified models (e.g., \(k\)-\(\varepsilon\) and \(k\)-\(\omega\) models), RSM (Reynolds Stress Model), and LES (LargeEddy Simulation). The RSM model, based on the concept of Reynolds-averaging, describes the evolution of turbulent flows by modeling the Reynolds stresses within the turbulence. While it can provide relatively accurate results for certain common turbulent flow problems, it oversimplifies the structure of turbulence and fails to capture detailed turbulent characteristics [26; 27]. Large-Eddy Simulation (LES) is a transient solution method that directly solves the motion of large-scale eddies. By using a filtering function to \"filter\" turbulence, eddies smaller than the filtering scale in the turbulence are \"filtered\" out to obtain equations that describe large eddies. The influence of small eddies on large eddies is reflected by introducing sub-grid-scale stress terms into the large-eddy equation. However, due to the enormous computational workload associated with the LES method, especially for High-Reynolds-number flows and complex geometries, current computational power cannot fully support Large-Eddy Simulations [28]. Among the two-equation eddy viscosity turbulence models, the \(k\)-\(\varepsilon\) model is effective in simulating fully developed turbulent flows far from solid boundaries, while the \(k\)-\(\omega\) model performs better in solving boundary layer problems under varying pressure gradients. [PERSON] combined the strengths of both models and proposed the SST (Shear Stress Transport) two-equation model [29]. This hybrid model has since been widely applied in engineering practice, which is why this study also selected the SST \(k\)-\(\omega\) model for numerical simulation. The governing equations are as follows:  
The transport equation for \(k\):  
\[\frac{\partial k}{\partial t}+\overline{U}_{j}\frac{\partial k}{\partial x_{j }}=\frac{\partial}{\partial x_{j}}[(\
u+\sigma_{k}\
u_{t})\frac{\partial k}{ \partial x_{j}}+\widetilde{P}_{k}-\beta^{*}k\omega] \tag{4}\]  
The transport equation for \(\omega\):  
\[\frac{\partial\omega}{\partial t}+\overline{U}_{j}\frac{\partial\omega}{ \partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[(\
u+\sigma_{\omega}\
u_ {t})\frac{\partial\omega}{\partial x_{j}}+\alpha S^{2}-\beta\omega^{2}+2(1-F_{ 1})\sigma_{\omega}\frac{1}{\omega}\frac{\partial k}{\partial x_{j}}\frac{ \partial\omega}{\partial x_{j}}\right] \tag{5}\]