# Assessment of Two Streamline Curvature Correction Methods for an Elliptic Blending Turbulence Model
## 2 Turbulence Modeling
After linearizing and transforming from the \(\varepsilon\)- to the \(\omega\)-form using the relation \(\varepsilon=\beta^{*}k\omega\), the regularized anisotropy tensor becomes  
\[b_{ij}=\frac{(3C_{2}-4)\big{(}1+\eta^{2}\big{)}}{(6+2\eta^{2}+12\zeta^{2}\eta ^{2}+12\zeta^{2})}\bigg{(}\frac{1}{\beta^{*}\omega}\bigg{)}gS_{ij}, \tag{7}\]  
where  
\[g=\bigg{(}\frac{1}{2}C_{1}+\frac{G_{k}^{\prime}}{\beta^{*}k\omega}-1\bigg{)}^ {-1}, \tag{8}\]  
\[\eta^{2}=S_{ij}^{*}S_{ij}^{*},\zeta^{2}=W_{ij}^{*}W_{ij}^{*}\, \tag{9}\]  
and  
\[S_{ij}^{*}=\frac{2-C_{3}}{2\beta^{*}\omega}gS_{ij},\ W_{ij}^{*}=\frac{2-C_{4}} {2\beta^{*}\omega}g\Omega_{ij}. \tag{10}\]  
The shear strain tensor and vorticity tensor are  
\[\frac{\partial\rho\omega}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_ {i}\omega)=f_{\omega}\gamma\frac{\omega}{k}F_{1}G_{k}-F_{2}\rho\beta\omega^{2 }+C_{D}+\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2}+\sigma_{ \omega}\mu_{t}\Big{)}\frac{\partial\omega}{\partial x_{j}}\Bigg{]}, \tag{2}\]0  
The production of turbulent kinetic energy, \(G_{k}^{\prime}\), is given as  
\[\frac{\partial\rho\omega}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_ {i}\omega)=f_{\omega}\gamma\frac{\omega}{k}F_{1}G_{k}-F_{2}\rho\beta\omega^{2 }+C_{D}+\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2}+\sigma_{ \omega}\mu_{t}\Big{)}\frac{\partial\omega}{\partial x_{j}}\Bigg{]}, \tag{2}\]1  
Using Equation (7), this becomes  
\[\frac{\partial\rho\omega}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_ {i}\omega)=f_{\omega}\gamma\frac{\omega}{k}F_{1}G_{k}-F_{2}\rho\beta\omega^{2 }+C_{D}+\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2}+\sigma_{ \omega}\mu_{t}\Big{)}\frac{\partial\omega}{\partial x_{j}}\Bigg{]}, \tag{2}\]2  
where \(S^{2}=2S_{ij}S_{ij}\). From conventional LEVMs, the production of turbulent kinetic energy (\(G_{k}\)) is given by  
\[\frac{\partial\rho\omega}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_ {i}\omega)=f_{\omega}\gamma\frac{\omega}{k}F_{1}G_{k}-F_{2}\rho\beta\omega^{2 }+C_{D}+\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2}+\sigma_{ \omega}\mu_{t}\Big{)}\frac{\partial\omega}{\partial x_{j}}\Bigg{]}, \tag{2}\]3  
If we define  
\[\frac{\partial\rho\omega}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_ {i}\omega)=f_{\omega}\gamma\frac{\omega}{k}F_{1}G_{k}-F_{2}\rho\beta\omega^{2 }+C_{D}+\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2}+\sigma_{ \omega}\mu_{t}\Big{)}\frac{\partial\omega}{\partial x_{j}}\Bigg{]}, \tag{2}\]4then Equation (13) becomes  
\[\frac{\partial\rho\omega}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_ {i}\omega)=f_{\omega}\gamma\frac{\omega}{k}F_{1}G_{k}-F_{2}\rho\beta\omega^{2 }+C_{D}+\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2}+\sigma_{ \omega}\mu_{t}\Big{)}\frac{\partial\omega}{\partial x_{j}}\Bigg{]}, \tag{2}\]5  
In the SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC2 model, the production of turbulent kinetic energy is calculated by Equation (16). This means that all \(G_{k}\) terms in the equations of the original SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model should be multiplied by the function \(F_{1}\). At the same time, \(F_{2}=1\). [PERSON] and [PERSON] [8] provided three sets of coefficients for their model. The set used in the present work is as follows:  
\[\frac{\partial\rho\omega}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_ {i}\omega)=f_{\omega}\gamma\frac{\omega}{k}F_{1}G_{k}-F_{2}\rho\beta\omega^{2 }+C_{D}+\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2}+\sigma_{ \omega}\mu_{t}\Big{)}\frac{\partial\omega}{\partial x_{j}}\Bigg{]}, \tag{2}\]6  
where \(\Pi_{b}=b_{\eta}b_{\eta}\). It should be emphasized that, in the original SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model, [PERSON]'s correction [17] was used for \(G_{k}\); namely, \(G_{k}=\mu_{t}S\Omega\). However, in the present work, Equation (14) is used for \(G_{k}\), as it was found that this produces better results.