## 2 Methodology
### Turbulence Model  
The turbulent flow field of the hydrodynamic torque converter was solved with RANS equations. The flow field of a hydrodynamic torque converter is considered as a mixture of vapor and liquid. The fluid in the flow field always maintains the conservation of mass and momentum, which can be expressed by the following equations.  
Continuity equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{j})}{\partial x_{j}}=0. \tag{1}\]  
Momentum equation:  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial(\rho u_{i}u_{j})}{ \partial x_{j}}=\rho f_{i}-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}[(\mu+\mu_{t})(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}-\frac{2}{3}\frac{\partial u_{k}}{\partial x_{k }}\delta_{ij})]. \tag{2}\]  
Mixture density:  
\[\rho=\alpha_{v}\rho_{v}+(1-\alpha_{v}\rho_{l}). \tag{3}\]  
In Equations (1) and (2), \(x_{i}\), \(u_{i}\), and \(f_{i}\) are the components of the displacement, velocity, and body force, respectively, that are in the \(i\) direction. \(\mu\) and \(\mu_{t}\) are the laminar and turbulent viscosities, respectively.  
The governing equations in the SST \(k\)-\(\omega\) model are for the turbulence kinetic energy \(k\) and specific dissipation rate \(\omega\):  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{ i}}=\frac{\partial}{\partial x_{i}}\bigg{[}(\mu+\sigma_{k}\mu_{t})\frac{ \partial k}{\partial x_{i}}\bigg{]}+P_{k}-\beta^{*}\rho k\omega \tag{4}\]  
\[\begin{array}{rl}\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho \omega u_{i})}{\partial x_{i}}=&\frac{\partial}{\partial x_{i}}\Big{[}(\mu+ \sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_{i}}\Big{]}-\beta\rho \omega^{2}\\ &+2(1-F_{1})\rho\omega_{w}\frac{1}{\omega}\frac{\partial k}{\partial x_{i}} \frac{\partial\omega}{\partial x_{i}}+\alpha\rho S^{2}\end{array} \tag{5}\]The blending function \(F_{1}\) is defined as  
\[\left\{\begin{array}{l}F_{1}{=}\tanh\left\{\left\{\min\left[max\left(\frac{ \sqrt{k}}{\beta^{*}\omega y^{*}},\frac{500v}{y^{*}\omega}\right),\frac{4\omega v \alpha_{2}k}{CD_{k}y^{2}}\right]\right\}\right\}^{4}\right\}\\ CD_{k\omega}=max\left(2\rho\omega_{2}\frac{1}{\omega}\frac{\partial k}{ \partial x_{i}}\frac{\partial k\omega}{\partial x_{i}},10^{-10}\right)\end{array}\right.. \tag{6}\]  
At the boundary layer, the \(k\)-\(\epsilon\) model is employed away from the wall. When \(F_{1}\) = 0, the inner layer switches to the \(k\)-\(\omega\) model. The turbulent viscosity is defined by  
\[\mu_{t}=\frac{a_{1}k}{max(a_{1}\omega,SF_{2})}. \tag{7}\]  
where \(S\) is the invariant measure of the strain rate. The second blending function \(F_{2}\) is defined as  
\[F_{2}=tanh\left[\left[max\left(\frac{2\sqrt{k}}{\beta^{*}\omega y},\frac{500v }{y^{2}\omega}\right)\right]^{2}\right]. \tag{8}\]  
In the stagnation zone, the SST \(k\)-\(\omega\) turbulence model defines a production limiter to prevent turbulence generation, which is defined as  
\[P_{k}=\mu_{t}\frac{\partial u_{i}}{\partial x_{j}}\left(\frac{\partial u_{i}} {\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right). \tag{9}\]  
Table 1 presents all constants in the SST \(k\)-\(\omega\) turbulence model.