# CFD Estimation of a Resistance Coefficient for an Egg-Shaped Geometric Dome
## 2 Materials and Methods  
### CFD Method  
The equations that describe the motion of fluid are called the Navier-Stokes equations and consist of two equations, the mass conservation equation [16]:  
\[\frac{\partial\rho}{\partial t}+\bigtriangledown\cdot(\rho\vec{u}\vec{t})=Q \tag{1}\]  
and the momentum conservation equation  
\[\frac{\partial}{\partial t}(\rho\vec{u})+\bigtriangledown\cdot(\rho\vec{u} \vec{u})=-\bigtriangledown p+\bigtriangledown(\vec{t})+\rho\vec{g}+\vec{S}, \tag{2}\]  
\[\vec{\tau}=\mu\bigg{[}\Big{(}\bigtriangledown\vec{u}+\bigtriangledown\vec{ u}^{T}\Big{)}-\frac{2}{3}\bigtriangledown\vec{u}\vec{I}\bigg{]}, \tag{3}\]where \(\rho\) is fluid density, \(p\) is pressure, \(u\) is fluid velocity, \(\tilde{\tau}\) is a stress tensor, \(\mu\) is fluid viscosity, \(I\) is unit tensor, \(\vec{g}\) is gravity, \(\vec{F}\) is external force, \(Q\) is the fluid source, and \(\vec{S}\) is the external momentum source. The solution of the N-S equations in general form, even numerically, would be computationally expensive for turbulent flows, which by definition is random by nature. Therefore, one of the widely used approaches in Computational Fluid Dynamics is to replace instantaneous variables with time-averaged (mean) values and their fluctuations. Any scalar quantities can be presented by  
\[\vec{\phi}=\phi+\phi^{\prime}, \tag{4}\]  
where \(\phi\) is a mean (time-averaged) value and \(\phi^{\prime}\) is the fluctuation over the mean value. This leads to the Reynolds Averaged N-S (RANS) equations in the form presented in the following. Assuming no mass transfer between fluid phases and no external forces act on the fluid, the source term in Equation (1) \(Q=0\) and the force \(\vec{S}\) in Equation (2) equals zero.The time averaged continuity equation in Cartesian coordinates is given by  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}(\rho u_{i})=0 \tag{5}\]  
and the time averaged momentum conservation equation is  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{i}}(\rho u _{i}u_{j})=\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x_{j}} \Bigg{[}\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}} {\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{j}}{\partial x_{i}} \right)\Bigg{]}+\frac{\partial}{\partial x_{j}}\Big{(}-\rho\overline{u_{i}{}^ {\prime}u_{j}}^{\prime}\Big{)}+\rho\vec{g}. \tag{6}\]  
The RANS equations remain open and the term \(-\rho\overline{u_{i}{}^{\prime}u_{j}}^{\prime}\), which is called Reynolds stress, must be modelled to close them. The most common approach is the Boussinesq approach, which defines the Reynolds stress to the gradient of mean fluid velocity gradient  
\[-\rho\overline{u_{i}{}^{\prime}u_{j}}^{\prime}=\mu_{t}\left(\frac{\partial u_ {i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3} \bigg{(}\rho k+\mu_{t}\frac{\partial u_{k}}{\partial x_{k}}\bigg{)}\delta_{ij}, \tag{7}\]  
where \(\delta_{ij}\) is the Kronecker delta:  
\[\delta_{ij}= \begin{cases}1&i=j\\ 0&i\
eq j\end{cases}  \tag{8}\]  
The turbulence models use the Boussinesq approach, among which the most commonly used is \(k-\epsilon\), which, however, has weaknesses in modeling separating and swirling flow and is dedicated mainly to high-Reynolds number flows. Another turbulent model that eliminates mentioned drawbacks is a two-equation turbulence model, the \(k-\omega\). It also has a few modifications, among which the Shear Stress Transport (\(SST\)) showed good agreement with flow over a cylinder, as it was proved in work [17]. This study uses the \(SST\) model to simulate flow over the egg-shaped tank.