# On the Accuracy of Turbulence Model Simulations of the Exhaust Manifold
## 2 Methodology
### Turbulence Models Equations
#### 2.3.1 The RNG K-\(\varepsilon\) Model  
The k-\(\varepsilon\) turbulence model is widely used in CFD simulations. It is a two-equation model based on the Reynolds-averaged Navier-Stokes (RANS) equations.  
The k-\(\varepsilon\) model consists of two transport equations: one is for turbulent kinetic energy (k) and one is for the dissipation rate of the turbulent kinetic energy (\(\varepsilon\)). The turbulent kinetic energy represents the kinetic energy associated with random fluctuations in velocity, while the dissipation rate represents the rate at which the turbulent kinetic energy is dissipated into thermal energy ([PERSON] et al. [24]).  
The k-\(\varepsilon\) model is widely recognized in the field of engineering for its simplicity and computational efficiency when dealing with turbulent flows. However, this model has been found to have limitations, particularly in the case of swirling flows, and its ability to accurately capture complex turbulent phenomena is questionable. As a result, advanced turbulence models, such as the large eddy simulation and direct numerical simulation, are sometimes employed to achieve more precise predictions. The judicious use of these models can lead to more accurate results in applications where precision is paramount ([PERSON] et al. [25] and [PERSON] et al. [26]):  
\[\frac{\partial}{\partial t}\left(\rho^{2}\right)+\frac{\partial}{\partial x_{ l}}\left(\rho^{2}x_{k}\right)=\frac{\partial}{\partial x_{j}}\left(\alpha_{k}\mu_{ kff}\frac{\partial k}{\partial x_{j}}\right)+G_{k}+G_{b}-Y_{K}-s_{k}-\rho_{k} \tag{8}\]  
and:  
\[=\frac{\partial}{\partial x_{j}}\left(\alpha_{s}\ \mu_{eff}\frac{\partial s}{ \partial x_{j}}\right)+C_{1}s\frac{\delta}{\xi}\left(G_{k}+C_{3}s_{G}\right)-C _{2}s\rho\frac{z}{\xi}^{2}-R_{s}+S_{s} \tag{9}\]  
The following terms are of significance when describing different factors that contribute to the generation of turbulent kinetic energy:  
* G\({}_{\rm k}\) signifies the production of turbulent kinetic energy resulting from mean velocity gradients.
* G\({}_{\rm b}\) denotes the generation of turbulent kinetic energy attributed to buoyancy.
* Y\({}_{\rm M}\) represents the involvement of fluctuating dilatation in compressible turbulence, contributing to the overall dissipation rate.
* \(\alpha_{\rm s}\) and \(\alpha_{\rm k}\) represent the reciprocal effective Prandtl numbers for \(\varepsilon\) and k, respectively.
* S\({}_{\rm k}\) and S\({}_{\rm s}\) are designated as user-defined source terms.  
The k-\(\varepsilon\) model assumes that the turbulent viscosity is proportional to the ratio of the turbulent kinetic energy to the dissipation rate. This relationship is expressed as:  
\[m_{ut}=\frac{\rho C_{mu}K^{2}}{\varepsilon} \tag{10}\]  
where \(m_{ut}\) is the turbulent viscosity, \(\rho\) is the density of the fluid, C\({}_{\rm mu}\) is a constant that is typically 0.09, k is the turbulent kinetic energy, and \(\varepsilon\) is the dissipation rate of turbulent kinetic energy.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Turbulence Model** & \
\begin{tabular}{c} **Number of** \\ **Equations** \\ \end{tabular} &
\begin{tabular}{c} **Transition** \\ **Modeling** \\ \end{tabular} & **Abbreviation** \\ \hline
1 & RNG k-\(\varepsilon\) model & 2 & No & RNG \\
2 & k-\(\omega\) model STD & 2 & No & RKE \\
3 & k-kl-\(\omega\) & 3 & YES & KKL \\
4 & SST transition & 4 & YES & TSST \\
5 & Reynolds stress & 7 & YES & \\ \hline \end{tabular}
\end{table}
Table 1: Turbulence models overview.