# Identification and Optimization Study of Cavitation in High Power Torque Converter
## 2 Torque Converter Simulation and Analytical Modeling
### Simulation Model of Torque Converter
The simulated equation for LES calculation after filtering is as follows:  
\[\frac{\partial\overline{u}_{i}}{\partial t}+\frac{\partial\overline{u}_{i} \overline{u}_{j}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{p}}{ \partial x_{i}}+\upsilon\frac{\partial^{2}\overline{u}_{i}}{\partial x_{i} \partial x_{j}}-\frac{\partial\tau_{ij}}{\partial x_{j}} \tag{3}\]  
\[\frac{\partial\rho}{\partial t}=-\frac{\partial x}{\partial x_{i}}(\rho \overline{u}_{i}) \tag{4}\]  
\(u_{i}\), \(u_{j}\) Time mean velocity in tensor, m/s. \(v\) Kinematic viscosity coefficient. \(p\) Fluctuation pressure. \(\tau_{ij}\) The sublattice stress term can be solved by the following equation. \[\left\{ \begin{array}{cc}\tau_{ij}=\frac{1}{3}\tau_{i}\delta_{ij}-2\mu_{i} \overline{\xi}_{ij}\\ \overline{S}_{ij}=\frac{1}{2}\Big{(}\frac{\partial\overline{u}_{i}}{\partial x _{j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\Big{)}\end{array} \right. \tag{5}\]  
\(\delta_{ij}\) Kronecker delta. \(\overline{S}_{ij}\) Stress tensor ratio. \(\tau_{r}\) Isotropic sublattice stress, Pa.  
Figure 2: Structural mesh model of the hydrodynamic torque converter. \(\mu_{i}\) Sublattice scale turbulent viscosity. FLUENT offers six sublattice models for the LES model. The Smagorinsky-Lilly (SL) sublattice model, which is currently more widely used, includes the expression for the turbulent viscous coefficient \(\mu_{t}\)[19]. \[\mu_{t}=\rho\Big{\{}min(0.41d;\ C_{S}V^{1/3})\Big{\}}^{2}\sqrt{2\overline{S_{ij} S_{ij}}} \tag{6}\]  
\(d\) Height of the first layer of the grid  
\(C_{S}\) SL sub-grid model constant  
\(V\) Volume of the computational cell  
The Shear stress transport (SST) model was employed to solve the boundary layer. When the turbulent scale \(L_{t}\) is smaller than the local grid size, the turbulence switches from the LES model to the SST model. The mixing term for the turbulent kinetic energy equation of the SST model \(\varepsilon_{SDES}\) is presented in the following equation [16];  
\[\left\{ \begin{matrix}\varepsilon_{SDES}=-\beta^{*}k\omega F_{SDES}\\ F_{SDES}=\Big{[}max\Big{(}\frac{L_{t}}{C_{DES}\Delta_{SDES}}(1-f_{s}),1\Big{)} -1\Big{]}\end{matrix} \right. \tag{7}\]  
\(\beta^{*}\) Model constants. \(k\) Turbulent kinetic energy. \(\omega\) Turbulence dissipation rate. \(F_{SDES}\) Correction factor. \(L_{t}\) Turbulent length. \(C_{DES}\) Simulation coefficient of shedding vortex,  
\(\Delta_{SDES}\) Grid characteristic length of shedding vortex. \(f_{s}\) Blending function. The blending function was applied to accelerate the transition from the RANS turbulence model to the LES turbulence model, which can be obtained by the following equation:  
\[\tau_{ij}^{SBES}=f_{s}\tau_{ij}^{RANS}+(1-f_{s})\tau_{ij}^{LES} \tag{8}\]  
\(\tau_{ij}^{SBES}\), \(\tau_{ij}^{RANS}\), \(\tau_{ij}^{LES}\) The stress tensor in stress-blended eddy simulation, RANS and LES turbulence mode  
The turbulence model and boundary condition settings for the flow field calculation are shown in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Parameter** \\ \hline Analysis Type & Transient Calculation \\ \hline Type of fluid domain & Full-flow channel model \\ \hline Turbulence model & RANS/LES \\ \hline Mesh type & Hexahedral structured mesh \\ \hline y+ value & \(\leq\)2 \\ \hline Density of hydraulic transmission oil & 860 kg/m\({}^{3}\) \\ \hline Fluid oil viscosity & 0.01784 kg/(m-s) \\ \hline Time step & 0.0005 s/step \\ \hline Simulation time step & 400 \\ \hline Pressure-velocity coupling method & SIMPLE \\ \hline Transient spatial discretization method & Bounded second-order implicit \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Turbulence modeling and boundary conditions for flow field calculations.