Turbulence Modeling

1 k-ω SST-SAS Equations

$\frac{\partial \omega}{\partial t}+\nabla \cdot (\vec{u}\omega) = \nabla^2[(\nu+\nu_t\sigma_\omega)\omega] + \frac{\gamma \omega}{k}\tau_{ij}\frac{\partial u_i}{\partial x_j}-\beta\omega^2 + 2(1-F_1)\sigma_{\omega2}\frac{1}{\omega}\nabla k \cdot \nabla \omega + P_{sas}$

$\frac{\partial k}{\partial t}+\nabla \cdot (\vec{u}k) = \nabla^2[(\nu+\nu_t\sigma_k)k] + \tau_{ij}\frac{\partial u_i}{\partial x_j}-\beta^*\omega k$

$P_{sas}=1.25 \mbox{max}\left(T_1-T_2,0\right)$

$T_1=1.755 \kappa S^2\frac{L}{L_{vK}}$

$T_2=3 k \mbox{max}\left(\frac{1}{\omega^2}\nabla \omega \cdot \nabla \omega , \frac{1}{k^2}\nabla k \cdot \nabla k \right)$

$L=\frac{k^{\frac{1}{2}}}{\omega c_\mu^{\frac{1}{4}}}$

$L_{vK}=\kappa \frac{S}{|\nabla^2 \vec{u}|}$

$\nu _T = {a_1 k \over \mbox{max}(a_1 \omega, \Omega F_2) }$

$F_2=\mbox{tanh} \left[ \left[ \mbox{max} \left( { 2 \sqrt{k} \over \beta^* \omega y } , { 500 \nu \over y^2 \omega } \right) \right]^2 \right]$

$F_1=\mbox{tanh} \left\{ \left\{ \mbox{min} \left[ \mbox{max} \left( {\sqrt{k} \over \beta ^* \omega y}, {500 \nu \over y^2 \omega} \right) , {4 \sigma_{\omega 2} k \over CD_{k\omega} y^2} \right] \right\} ^4 \right\}$

$CD_{k\omega}=\mbox{max} \left( 2\rho\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right )$

φ = φ1F1 + φ2(1 − F1)

$\alpha_1 = {{5} \over {9}}, \alpha_2 = 0.44$

$\beta_1 = {{3} \over {40}}, \beta_2 = 0.0828$

$\beta^* = {9 \over {100}}$

σk1 = 0.85,σk2 = 1

σω1 = 0.5,σω2 = 0.856

2 Solver

The openFoam solver is a modified version of turbFoam. Since the k-ω SST sas turbulence model, in unstable flow situations, gives LES-like solution fields, the modifications were introduced to handle the omega field and to allow the time averaging of pressure and velocity quantities. The averaging has been done in analogy with the openFoam LES solver oodles.

The solver files are available here: Media:turbFoamAverage.tar.gz

3 Turbulence Model Files

Note: This library was obviously developed with 1.4.1 or older and won't compile with 1.5 or newer The turbulence models files are available here: Media:KOmegaTurbulenceModels.tar.gz

4 Test Cases

4.1 Flow Past a Long Square Cylinder

Details about the test case are available at the ERCOFTAC NEXUS data base (ErcoftacLESig)

Test case files are available here:Media:LESigSquareCylSSTsas.tar.gz