# Numerical Study on Tip Vortex Cavitation Inception on a Foil
## 2 Numerical Approaches  
### Governing Equations of Flows  
The tip vortex flows around an elliptic hydrofoil were described by the RANS equations in which the flow quantities are split into an average and a fluctuating part. They can be derived from mass and momentum conservation principles based on the assumption of continuum mechanics. The continuity and momentum equations for two-phase incompressible flows in differential conservation form can be written as follows:  
\[\frac{\partial\overline{u}_{i}}{\partial x_{j}}=0 \tag{1}\]  
\[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial(\rho \overline{u}_{i}\overline{u}_{j})}{\partial x_{j}}=-\frac{\partial\overline{p} }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left(\mu\frac{\partial \overline{u}_{i}}{\partial x_{j}}-\rho\overline{u^{\prime}_{i}u^{\prime}_{j}}\right) \tag{2}\]  
where \(\rho\) is the density, \(t\) is time, \(u_{i}\) are the velocity components in the direction of the Cartesian coordinates \(x_{i}\), \(u^{\prime}_{i}\) are the velocity fluctuations, \(p\) is the pressure, and \(\mu\) is the dynamic viscosity. The unclosed term \(\tau_{ij}=\rho\overline{u^{\prime}_{i}u^{\prime}_{j}}\) is the turbulent Reynolds stress tensor. Most turbulence models rely on the so-called Boussinesq analogy, whereby the anisotropic part of \(\tau_{ij}\) is written as a linear function of the mean rate of strain as follows:  
\[\tau_{ij}=-\mu_{t}\left(\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac {\partial\overline{u}_{j}}{\partial x_{i}}\right)+\frac{2k}{3}\delta_{ij} \tag{3}\]  
where \(k\) is the turbulent kinetic energy, \(\frac{2}{3}k\delta_{ij}\) in the isotropic part of the Reynolds stresses, and \(\mu_{t}\) is the turbulent eddy viscosity. On the other hand, the Reynolds stresses can be determined by solving the Reynolds stress transport equations as follows:  
\[ \begin{array}{l}\frac{\partial}{\partial t}\left(\rho\overline{u^{\prime}_{ i}u^{\prime}_{j}}\right)+\frac{\partial}{\partial x_{k}}\left(\rho u_{k} \overline{u^{\prime}_{i}u^{\prime}_{j}}\right)=-\frac{\partial}{\partial x_{k }}\left[\rho\overline{u^{\prime}_{i}u^{\prime}_{i}u^{\prime}_{k}}+\overline{p \left(\delta_{kj}u^{\prime}_{i}+\delta_{ik}u^{\prime}_{j}\right)}\right]\\ +\frac{\partial}{\partial x_{k}}\left[\mu\frac{\partial}{\partial x_{k}} \left(\overline{u^{\prime}_{i}u^{\prime}_{j}}\right)\right]-\rho\left( \overline{u^{\prime}_{i}u^{\prime}_{j}}\frac{\partial u_{j}}{\partial x_{k}} +\overline{u^{\prime}_{i}u^{\prime}_{k}}\frac{\partial u_{j}}{\partial x_{k}} \right)-\rho\beta\left(g\overline{u^{\prime}_{i}\theta}+g\overline{u^{\prime} _{i}\theta}\right)\\ +p\left(\frac{\partial u^{\prime}_{i}}{\partial x_{j}}+\frac{\partial u^{ \prime}_{j}}{\partial x_{k}}\right)-2\mu\frac{\partial u^{\prime}_{i}}{ \partial x_{k}}\frac{\partial u^{\prime}_{j}}{\partial x_{k}}-2\rho\Omega_{k }\left(\overline{u^{\prime}_{i}u^{\prime}_{m}}\epsilon_{ikm}+\overline{u^{ \prime}_{i}u^{\prime}_{m}}\epsilon_{jkm}\right)\end{array}  \tag{4}\]  
where \(\beta\) is the coefficient of expansion, \(g_{i}\) are the gravity accelerations, and \(\Omega_{k}\) are the rotation rates. The seven terms on the right-hand side of Equation (4) are interpreted as turbulence diffusion, molecular diffusion, stress production, buoyancy production, pressure-strain and dissipation from left to right. This kind of turbulence modeling is referred to as the Reynolds stress model (RSM) which was originally discussed by [PERSON] et al. (2016). Different turbulence models were selected in the simulations of this paper. Among several two-equation models, the realizable \(k\)-\(\varepsilon\) model based on the suggestions by [PERSON] et al. (2017) and the shear stress transport \(k\)-\(\omega\) turbulence model of [PERSON] (2018) were used, where abbreviations for the former and the latter model in this paper are RKE and SKW, respectively. The RKE model uses the same turbulence viscosity assumption as the standard \(k\)-\(\varepsilon\) model but substantially shows better performances for many applications.