# CFD Design Performance Analysis for a High-Speed Propeller
## 2 Theoretical Considerations  
One of the best turbulence models was developed by dr. [PERSON] and presented since 1994-Shear Stress Transport (SST) turbulence model [19; 20]. This model is based on physical experiments, and consists of a hybrid formulation, strategically combining the strengths of k-\(\omega\) and k-\(\epsilon\) models, reducing their respective limitations. [PERSON] introduced a blending mechanism between these two models using a function, F\({}_{1}\) (see Equation (2)) which switches between the models' equations. The approach of this model is to solve two transport equations: for the turbulent kinetic energy, \(k\left[\frac{\text{m}^{2}}{\text{s}^{2}}\right]\), and for the specific dissipation rate, \(\omega\left[\frac{1}{\text{s}}\right]\)[21; 22]. 1. The equation for turbulent kinetic energy, \(k\) \[\frac{\partial(pk)}{\partial t}+\frac{\partial\left(pu_{i}k\right)}{\partial x _{j}}=P_{k}-\rho\beta^{*}k\omega+\frac{\partial}{\partial x_{j}}\left[(\mu+ \sigma_{k}\mu_{t})\frac{\partial k}{\partial x_{j}}\right],\] (1) where \(\rho\) is density \(\left[\text{kg}/\text{m}^{3}\right]\); \(u_{i}\) is the mean velocity component \(\left[\text{m}/\text{s}\right]\); \(\text{P}_{k}\) is the production of turbulent kinetic energy \(\left[\frac{\text{kg}}{\text{m}\cdot\text{s}^{3}}\right]\); \(u_{t}\) is eddy viscosity \(\left[\frac{\text{kg}}{\text{m}\cdot\text{s}}\right]\); \(\beta^{*}\) and \(\sigma_{k}\) are model constants, dimensionless. In Equation (1) the left-hand side terms represent the temporal evolution and convective transport of \(k\). On the right-hand side, \(P_{k}\) signifies the production of turbulent kinetic energy, which is derived from the interaction of mean velocity gradients with the Reynolds stresses. The term \(-\rho\beta^{*}k\omega\) quantifies the rate of dissipation of turbulent kinetic energy into thermal energy. The final term describes the turbulent diffusion of \(k\), influenced by both molecular viscosity \((\mu)\) and the turbulent eddy viscosity \((\mu_{t})\), \((\sigma_{k})\) being a model constant. 1. The transport equation for specific dissipation rate, \(\omega\)  
This equation is more complex due to the blending mechanism  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial\left(pu_{i}\omega \right)}{\partial x_{j}}=\alpha\frac{\omega}{k}P_{k}-\rho\beta\omega^{2}+ \frac{\partial}{\partial x_{j}}\left[(\mu+\sigma_{\omega}\mu_{t})\frac{ \partial\omega}{\partial x_{j}}\right]+2\rho(1-F_{1})\sigma_{\omega 2}\frac{1}{ \omega}\frac{\partial k}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}, \tag{2}\]  
In Equation (2), \(\alpha\), \(\beta\) and \(\sigma\omega\) are model constants, dimensionless. Like in the \(k\) equation, the left part of the equation accounts for transient and convective transport. The right part of the equation includes the following:  
* Production: \(\alpha\frac{\omega}{k}P_{k}\)
* Dissipation: \(\rho\beta\omega^{2}\)
* Diffusion: \(\frac{\partial}{\partial x_{j}}\left[(\mu+\sigma_{\omega}\mu_{t})\frac{ \partial\omega}{\partial x_{j}}\right]\)  
The most important element of this equation is the cross-diffusion term \(2\rho(1-F_{1})\sigma_{\omega 2}\frac{1}{\omega}\frac{2\omega}{\partial x_{j}} \frac{\partial\omega}{\partial x_{j}}\). This term, weighted by \((1-F_{1})\) becomes active predominantly in the k-\(\epsilon\) region \((\)where \(F_{1}\approx 1)\) transforming the k-\(\omega\) equation into k-\(\epsilon\) and ensuring consistency in the far field.