# Contrib/turbFoamAverage with k-omega SST SAS

## 1 k- $\omega$ SST-SAS Equations $\frac{\partial \omega}{\partial t}+\nabla \cdot (\vec{u}\omega) = \nabla^2[(\nu+\nu_t\sigma_\omega)\omega] + \frac{\gamma \omega}{k}\tau_{ij}\frac{\partial u_i}{\partial x_j}-\beta\omega^2 + 2(1-F_1)\sigma_{\omega2}\frac{1}{\omega}\nabla k \cdot \nabla \omega + P_{sas}$ $\frac{\partial k}{\partial t}+\nabla \cdot (\vec{u}k) = \nabla^2[(\nu+\nu_t\sigma_k)k] + \tau_{ij}\frac{\partial u_i}{\partial x_j}-\beta^*\omega k$ $P_{sas}=1.25 \mbox{max}\left(T_1-T_2,0\right)$ $T_1=1.755 \kappa S^2\frac{L}{L_{vK}}$ $T_2=3 k \mbox{max}\left(\frac{1}{\omega^2}\nabla \omega \cdot \nabla \omega , \frac{1}{k^2}\nabla k \cdot \nabla k \right)$ $L=\frac{k^{\frac{1}{2}}}{\omega c_\mu^{\frac{1}{4}}}$ $L_{vK}=\kappa \frac{S}{|\nabla^2 \vec{u}|}$ $\nu _T = {a_1 k \over \mbox{max}(a_1 \omega, \Omega F_2) }$ $F_2=\mbox{tanh} \left[ \left[ \mbox{max} \left( { 2 \sqrt{k} \over \beta^* \omega y } , { 500 \nu \over y^2 \omega } \right) \right]^2 \right]$ $F_1=\mbox{tanh} \left\{ \left\{ \mbox{min} \left[ \mbox{max} \left( {\sqrt{k} \over \beta ^* \omega y}, {500 \nu \over y^2 \omega} \right) , {4 \sigma_{\omega 2} k \over CD_{k\omega} y^2} \right] \right\} ^4 \right\}$ $CD_{k\omega}=\mbox{max} \left( 2\rho\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right )$ $\phi = \phi_1 F_1 + \phi_2 (1 - F_1)$ $\alpha_1 = {{5} \over {9}}, \alpha_2 = 0.44$ $\beta_1 = {{3} \over {40}}, \beta_2 = 0.0828$ $\beta^* = {9 \over {100}}$ $\sigma_{k1} = 0.85, \sigma_{k2} = 1$ $\sigma_{\omega 1} = 0.5, \sigma_{\omega 2} = 0.856$