# Explicit nonlinear constitutive relation

 Turbulence RANS-based turbulence models Linear eddy viscosity models Nonlinear eddy viscosity models Explicit nonlinear constitutive relation v2-f models $\overline{\upsilon^2}-f$ model $\zeta-f$ model Reynolds stress model (RSM) Large eddy simulation (LES) Detached eddy simulation (DES) Direct numerical simulation (DNS) Turbulence near-wall modeling Turbulence free-stream boundary conditions
\begin{align} - \frac{\mathbf{u u}}{k} & + \frac{2}{3} \mathbf{I} = \beta_1 \tilde{\mathbf{S}} \\ & + \beta_2 \left( \tilde{\mathbf{S}}^2 - \frac{II_S}{3} \mathbf{I} \right) + \beta_3 \left( \tilde{\mathbf{\Omega}}^2 - \frac{II_\Omega}{3} \mathbf{I} \right) \\ & + \beta_4 \left( \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \right) + \beta_5 \left( \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 \right) \\ & + \beta_6 \left( \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}} - \frac{2}{3} IV \mathbf{I} \right) \\ & + \beta_7 \left( \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}}^2 - \frac{2}{3} V \mathbf{I} \right) \\ & + \beta_8 \left( \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 + \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \right) + \beta_9 \left( \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} \right) \\ & + \beta_{10} \left( \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} \right) \end{align}