# Identification and Optimization Study of Cavitation in High Power Torque Converter
## 2 Torque Converter Simulation and Analytical Modeling
### Simulation Model of Torque Converter  
The simulation analysis of hydraulic torque converters involves two calculation models: a two-phase flow model that considers cavitation and a single-phase flow model that does not. The latter can be used to calculate the flow field of a unidirectional flow model without cavitation. The absolute pressure in the results is analyzed to determine whether cavitation occurs or not [16]. For this study, we utilized the impeller-closed flow field model without cavitation and entrance/exit setup. To simplify the model, we ignored any liquid leakage between each impeller and the casting fillet. The full flow channel of the impeller is established, as depicted in Figure 1.  
The hexahedral meshing technique is utilized for meshing. To enhance the accuracy of the calculation, boundary layer meshing is applied to refine the near-wall meshes of the pump, turbine, and stator. The height of the first layer mesh was set to 0.0235 mm, with a growth coefficient of 1.25 and a cumulative increase of 10 layers, ensuring that the average y+ value was <2. CFD models with different grid sizes were established to balance the relationship between computational effort and accuracy. Numerical simulations were conducted in stall condition and the zero-speed nominal torque of the pump was extracted for analysis. As shown in Table 2, when the grid quantity increased from 7.48 million to 8.38 million, the calculation error was controlled within 0.3%, while the calculation time significantly increased from 6.5 h to 11.4 h. In consideration of the computational cost and prediction accuracy, the 7.48 million grid cells were employed in this paper. The grid model is illustrated in Figure 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Tot. Elements (\(\times\)10\({}^{6}\))** & \(M_{\text{gg0}}\) (N\(\cdot\)m) & **Div./\%** & **Duration (h)** \\ \hline
2.59 & 796.9 & - & 1.8 \\ \hline
3.77 & 876.9 & 10.04\% & 2.5 \\ \hline
4.38 & 949.7 & 8.30\% & 3.3 \\ \hline
5.31 & 962.5 & 1.35\% & 3.9 \\ \hline
7.48 & 966.2 & 0.38\% & 6.5 \\ \hline
8.38 & 968.3 & 0.22\% & 11.4 \\ \hline \end{tabular}
\end{table} 
Table 2: Grid independence analysis. Figure 1: Torque converter CFD modeling. The Reynolds-averaged Navier-Stokes equation (RANS) turbulence model and large eddy simulation (LES) are currently the most commonly used turbulence models. RANS is primarily based on the Reynolds time-averaged Navier-Stokes (N-S) equations for solving turbulence problems. It introduces the Reynolds stress term, which is generated by turbulent pulsation velocity in the original N-S equation, providing high accuracy in solving near-wall turbulent motion. On the other hand, LES operates by first spatially filtering the N-S equation and directly simulating large-scale eddies in turbulence using the instantaneous N-S equation. The impact of small eddies on large eddies is accounted for through an approximate sub-grid model. In recent years, hybrid models have been developed, employing the RANS method to solve turbulent boundary layer flow in the near wall region and the LES method in the far-away wall region to enhance computational accuracy [17; 18]. The control equation for LES is based on grid size screening and obtained by filtering the N-S function. The filter function is as follows:  
\[G\big{(}|x-x^{\prime}|\big{)}=\left\{ \begin{array}{cc}0&|x_{i}-x^{\prime}_{i }|>\frac{\Delta x_{i}}{2}(i=1,2,3)\\ \frac{1}{\Delta x_{1}\Delta x_{2}\Delta x_{3}}|x_{i}-x^{\prime}_{i}|\leq\frac {\Delta x_{i}}{2}(i=1,2,3)\end{array} \right. \tag{2}\]  
\(G\) Filter function. \(\Delta x_{i}\) The grid scale along the i direction in the coordinate system.