# Analyzing the Biomechanical Characteristics of Ski Jumping Take-Off Phase Based on CFD
## 2 Method
### Fluid Dynamics Calculations
#### 2.4.3 Governing Equations  
This study examines the aerodynamic characteristics of athletes using the k-\(\epsilon\) model, a widely adopted approach for fluid turbulence calculations, with its validity supported by previous verification studies [21]. This model is grounded in the Reynolds-Averaged Navier-Stokes (RANS) equations and incorporates two additional equations to close the turbulent stress calculation: one for turbulent kinetic energy (k) and another for the turbulent dissipation rate (\(\epsilon\)). The governing equations are as follows:  
\[\frac{\partial(\rho\mathrm{k})}{\partial\mathrm{t}}+\frac{\partial(\rho \mathrm{ku}_{\mathrm{i}})}{\partial\mathrm{x}_{\mathrm{i}}}=\frac{\partial}{ \partial\mathrm{x}_{\mathrm{i}}}\bigg{[}\bigg{(}\mu+\frac{\mu_{\mathrm{t}}}{ \sigma_{\mathrm{k}}}\bigg{)}\frac{\partial\mathrm{k}}{\partial\mathrm{x}_{ \mathrm{i}}}\bigg{]}+\mathrm{P}_{\mathrm{k}}-\rho\epsilon \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Elite Chinese/NW Athletes**} & \multicolumn{1}{c}{\begin{tabular}{c} **Ankle Joint Angle** \\ **(Mean \(\pm\) SD)** \\ \end{tabular} } & \begin{tabular}{c} **Knee Joint Angle** \\ **(Mean \(\pm\) SD)** \\ \end{tabular} & \begin{tabular}{c} **Hip Joint Angle** \\ **(Mean \(\pm\) SD)** \\ \end{tabular} \\ \hline SMT & \begin{tabular}{c} CA \\ NWA \\ \end{tabular} & \begin{tabular}{c} 61.2 \(\pm\) 13.8 \\ 40.69 \(\pm\) 3.67 \\ \end{tabular} & \begin{tabular}{c} 90.9 \(\pm\) 19.3 \\ 95.45 \(\pm\) 6.91 \\ \end{tabular} & \begin{tabular}{c} 61.1 \(\pm\) 20 \\ 60.64 \(\pm\) 7.36 \\ \end{tabular} \\ \hline EMT & \begin{tabular}{c} CA \\ NWA \\ \end{tabular} & \begin{tabular}{c} 87.9 \(\pm\) 8.37 \\ 56.14 \(\pm\) 5.56 \\ \end{tabular} & \
\begin{tabular}{c} 151.19 \(\pm\) 11.56 \\ 180.93 \(\pm\) 11.86 \\ \end{tabular} &
\begin{tabular}{c} 133.09 \(\pm\) 14.7 \\ 149.51 \(\pm\) 10.95 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Kinematic data of athletes at the start and end of the take-off phase. Figure 3: Schematic diagram of the computational region. Equation (5) is the k-equation (turbulent kinetic energy equation), which primarily describes the transport and generation of turbulent kinetic energy (k). Here, \(\rho\) represents the fluid density; \(\upmu\) is the velocity component; \(\upmu\) and \(\upmu\) are spatial coordinates; \(\upmu\) is the molecular viscosity coefficient; \(\mu\)t is the turbulent viscosity coefficient; \(\partial\)k is the turbulent Prandtl number for the k-equation (typically set to 1.0); \(\mathrm{P_{k}}\) represents the production of turbulent kinetic energy; and \(\epsilon\) denotes the dissipation rate of turbulent kinetic energy. \[\frac{\partial(\rho\epsilon)}{\partial\up t}+\frac{\partial(\rho\epsilon \upmu_{\mathrm{i}})}{\partial\up x_{\mathrm{i}}}=\frac{\partial}{\partial \up x_{\mathrm{j}}}[(\upmu+\frac{\upmu_{\mathrm{t}}}{\sigma_{\epsilon}}) \frac{\partial\epsilon}{\partial\up x_{\mathrm{j}}}]+\mathrm{C_{1}}\epsilon \frac{\epsilon}{\mathrm{k}}\mathrm{P_{k}}-\mathrm{C_{2}}\epsilon\rho\frac{ \epsilon^{2}}{\mathrm{k}} \tag{6}\]  
Equation (6) represents the \(\epsilon\)-equation (turbulent kinetic energy dissipation rate equation), which describes the transport and generation of the turbulent kinetic energy dissipation rate \(\epsilon\). \(\partial\epsilon\) is the turbulent Prandtl number for the \(\epsilon\)-equation (typically set to 1.3), and \(\mathrm{C_{1}}\epsilon\) and \(\mathrm{C_{2}}\epsilon\) are model constants, commonly taken as 1.44 and 1.92, respectively. In studying the aerodynamic characteristics of athletes using the k-\(\epsilon\) model, the continuity equation ensures mass conservation throughout the analysis. The Navier-Stokes equation describes momentum conservation, while the k-\(\epsilon\) model serves to close the turbulent stress term in the Navier-Stokes equation. In this study, these three equations jointly constitute a complete set of fluid dynamics equations, which are used to describe the behavior of fluids during flow.