# Examination of Viscosity Effect on Cavitating Flow inside Poppet Valves Based on a Numerical Study
## 2 Materials and Methods  
The employed numerical method involves the VOF (volume of fluid) algorithm [16] with incorporation of an extra cavitation model, which is responsible for prediction of cavitation evolution. Since two phases are present inside the poppet valve, only one single phase transport equation is required to represent the cavitating flow. \[\frac{\partial\alpha_{l}}{\partial t}+\
abla\cdot(\alpha_{l}U)+\
abla\cdot[ \alpha_{l}\alpha_{v}U_{r}]=\alpha_{l}\alpha_{v}\left[\frac{\rho_{v}}{\rho_{v} }-\frac{\dot{\rho}_{l}}{\rho_{l}}\right]+\alpha_{l}[\
abla\cdot U]+\dot{m} \left[\frac{1}{\rho_{l}}-\alpha_{l}\left(\frac{1}{\rho_{l}}-\frac{1}{\rho_{v}} \right)\right], \tag{1}\]  
The third term on the left-hand side of Equation (1) consists of relative velocity between the liquid and vapor phases indicated by Ur, for the sake of achieving a thin interface between phases [16]. The expression for such relative velocity is given by  
\[U_{r}=c_{a}|U|\frac{\
abla\alpha}{|\
abla\alpha|}, \tag{2}\]  
The local velocity is determined by pressure, viscous shearing effect and surface tension, given by  
\[\frac{\partial\rho U}{\partial t}+\
abla\cdot(\rho U\otimes U)=-\
abla p+ \
abla\cdot\tau+\sigma\kappa\
abla\alpha, \tag{3}\]  
The third term on the right-hand side of Equation (3) is used to model the surface tension effect, and the curvature of interface is approximated with the phase gradient as  
\[\kappa=\
abla\cdot\left(\frac{\
abla\alpha}{|\
abla\alpha|}\right), \tag{4}\]  
Establishment of the pressure equation [17] first involves a discretization of the momentum equation. \[U=\frac{H(U)}{a_{p}}-\frac{1}{a_{p}}\
abla p, \tag{5}\]A divergence procedure is performed to Equation (5). In conjunction with the velocity divergence obtained from the phase transport equations, the following pressure equation is derived. \[-\left(\frac{\alpha_{l}}{\rho_{l}}\bigg{(}\frac{D\rho_{l}}{Dt}\bigg{)}+\frac{ \alpha_{v}}{\rho_{v}}\bigg{(}\frac{D\rho_{v}}{Dt}\bigg{)}\right)+\dot{m}\bigg{(} \frac{1}{\rho_{l}}-\frac{1}{\rho_{v}}\bigg{)}=\bigg{(}\
abla\cdot\bigg{(}\frac {H(U)}{a_{p}}\bigg{)}-\
abla\cdot\bigg{(}\frac{1}{a_{p}}\
abla p\bigg{)}\bigg{)}, \tag{6}\]  
The compressibility of both phases in the pressure equation assumes an implicit treatment to enable the transonic calculation of the mixture region. \[-\left(\frac{\alpha_{l}}{\rho_{l}}\bigg{(}\frac{\dot{\varphi}_{l} \partial p}{\partial t}+U\cdot\
abla(\psi_{l}p)\bigg{)}+\frac{\alpha_{v}}{ \rho_{v}}\bigg{(}\frac{\dot{\varphi}_{v}\partial p_{v}}{\partial t}+U\cdot \
abla(\psi_{v}p)\bigg{)}\right)+\dot{m}\Big{(}\frac{1}{\rho_{l}}-\frac{1}{ \rho_{v}}\Big{)}=\] \[\left(\
abla\bigg{(}\frac{H(U)}{a_{p}}\bigg{)}-\
abla\cdot\left( \frac{1}{a_{p}}\
abla p\bigg{)}\right)\]  
where, the perfect fluid assumption is applied. Compressibility assumes a constant value. \[\rho=\psi p, \tag{8}\]  
The Schnerr-Sauer model, derived from generalized Rayleigh-Plesset equation [18], is applied for evaluation of interphase mass transfer. \[\left\{ \begin{array}{l}\dot{m}^{-}=\frac{3\varphi\rho_{v}}{\rho}\alpha_{l} \alpha_{v}(rR_{b})\sqrt{\min(p-p_{v},0)}\\ \dot{m}^{+}=\frac{3\varphi\rho_{v}}{\rho}\alpha_{l}\alpha_{v}(rR_{b})\sqrt{ \max(p_{v}-p,0)}\end{array} \right., \tag{9}\]  
In the listed equations, the subscript \(l\) and \(v\) indicate water phase and vapor phase, respectively. The designation of the variables in the mathematical model is provided in Table 1. More details of the numerical method could be found in a previous study [13]. The mathematical model is implemented on open-source platform OpenFOAM. Crank-Nicolson method with two-order accuracy is used for discretization of the temporal terms, while the divergence term utilizes the limited Linear method with a factor of 1. The phase transport equation is solved with MULES algorithm, and time advancement is addressed with PIMPLE loop, a combination of PISO and SIMPLE method.