# Analysis of the Guide Vane Jet-Vortex Flow and the Induced Noise in a Prototype Pump-Turbine
## 3 Mathematical Modeling
### Turbulence Modeling  
To close the RANS equations, eddy viscosity \(\mu_{t}\) can be used to build the relationship between Reynolds stress \(\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\) and the mean rate of strain tensor by:  
\[-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}=2\mu_{t}\overline{S_{ij}}-\frac{ 2}{3}k\delta_{ij} \tag{7}\]  
where \(k\) is the turbulence kinetic energy. The shear stress transport model based Detached Eddy Simulation (SST-DES) method is used for turbulent flow modeling. It is a zonal hybrid method [34] based on the Reynolds-averaged Navier-Stokes (RANS) equations and Large-Eddy Simulation (LES) equations. The SST model is used as the turbulence model for the time-averaged equations [35]. In the SST model, the turbulence kinetic energy \(k\) equation and specific dissipation rate \(\omega\) equation can be specified as:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{i}k)}{\partial x_{i }}\ =\ P-\frac{\rho k^{3/2}}{l_{k-\omega}}+\frac{\partial}{\partial x_{i}}\left[( \mu+\sigma_{k}\mu_{t})\frac{\partial k}{\partial x_{i}}\right] \tag{8}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho u_{i}\omega)}{ \partial x_{i}}\ =\ C_{\omega}P-\beta\rho\omega^{2}+\frac{\partial}{\partial x_{i}}\left[( \mu_{l}+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_{i}}\right]+ 2(1-F_{1})\frac{\rho\sigma_{\omega 2}}{\omega}\frac{\partial k}{\partial x_{i}}\frac{ \partial\omega}{\partial x_{i}} \tag{9}\]  
where \(l_{k-\omega}\) is the turbulence scale, which can be expressed as:  
\[l_{k-\omega}\ =\ k^{1/2}\beta_{k}\omega, \tag{10}\]  
in which \(P\) is the production term, \(C_{\omega}\) is the coefficient of the production term, \(F_{1}\) is the blending function, and \(\sigma_{k}\), \(\sigma_{\omega}\), and \(\beta_{k}\) are the model constants. In the DES method, the term \(\min(l_{k-\omega}\), \(C_{DES}\Delta_{m})\) is used instead of simple \(l_{k-\omega}\) term where \(\Delta_{m}=\max(\Delta x\), \(\Delta y\), \(\Delta z)\) is the mesh length scale, which is the maximum side length of an mesh element. If \(l_{k-\omega}\)\(<\)\(C_{DES}\Delta\), the LES model will be used instead of the RANS model for modeling the turbulence flow field.  
As widely known, turbulent flow in turbomachinery is mainly anisotropic. The SST model and other eddy viscosity models are based on the turbulence isotropic assumption. It can be somehow correct in engineering cases, but it lacks accuracy, especially in modeling local secondary flows. Corrections or improvements can be supplied for eddy viscosity models to have a better simulation. Therefore, DES is used to have a better solution of the jet and vortex flow field in this study. It may improve the simulation mainly in the large scale eddies. By predicting the Reynolds number (about \(2.8\times 10^{6}\)) and approximately calculating the upstream freestream velocity at the draft tube inlet (about1.2 m/s), one is able to solve out the viscous length scale (about 0.017 mm). On one hand, the largest longitudinal eddy scale is about 30-300 times the viscous length scale. It is from about 0.5 mm to about 5 mm. On the other hand, the eddy scale of turbulence kinetic energy dissipation is about 4-8 times of the viscous length scale. It is about 0.066-0.132 mm. Considering the balance computation cost and accuracy, the DES model is mainly used for resolving the 0.5-5 mm eddies by applying a suitable mesh length scale \(\Delta_{m}\). The effectiveness of DES will be checked and discussed after analyzing the results.