kOmegaTNT model
The kOmegaTNT model is a simple two-equation RANS turbulence model developed by J. C. Kok (Kok 2000). It is a modification of the original Wilcox k-ω model (Wilcox 1994), designed to remove strong dependency on the free-stream value of ω. This is achieved by including a cross-diffusion term in the ω equation, similar to the SST model by Menter (Menter 1994), and by re-calibration of model coefficients.
Model Equations
The transport equations for the kOmegaTNT model are given by:
\[ \begin{aligned} \frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho k u_j)}{\partial x_j} &= P_k - \beta^* \rho k \omega + \frac{\partial}{\partial x_j} \left[ (\mu + \sigma_k \mu_t) \frac{\partial k}{\partial x_j} \right], \\ \frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho \omega u_j)}{\partial x_j} &= P_\omega - \beta \rho \omega^2 + \frac{\partial}{\partial x_j} \left[ (\mu + \sigma_\omega \mu_t) \frac{\partial \omega}{\partial x_j} \right] + C_D, \end{aligned} \] where: \[ \begin{aligned} P_k &= \tau_{ij} \frac{\partial u_i}{\partial x_j}, \\ P_\omega &= \frac{\alpha_\omega \omega}{k} P_k, \\ C_D &= \frac{\sigma_d \rho}{\omega} \max\left[\frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}, 0\right]. \end{aligned} \]
Turbulent viscosity is computed as: \[ \mu_t = \rho \frac{k}{\omega}. \]
Model Coefficients
The model coefficients are set to the following values:
- \(\beta^* = 0.09\)
- \(\beta = 0.075\)
- \(\sigma_k = \frac{2}{3}\)
- \(\sigma_\omega = 0.5\)
- \(\alpha_\omega = 5/9\)
- \(\sigma_d = 0.5\)
Modifications
There are two optional modifications available for the kOmegaTNT model:
production limiter: Limits the production term \(P_k\) to prevent excessive turbulence generation in adverse pressure gradient flows. The limiter is applied to the equation for \(k\) where it replaces \(P_k\) with: \[ \tilde{P}_k = \min\left(P_k, 20 \beta^* \rho k \omega\right). \]
shock limiter: Reduces the turbulent viscosity in the presence of shocks to avoid nonphysical results. The limiter is applied to the equation for \(k\) where it replaces \(P_k\) with: \[ \tilde{P}_k = \min\left(P_k, k \sqrt{S_{ij} S_{ij}}\right). \]
The source term in the \(\omega\) equation is left unmodified.
Notes on OpenFOAM implementation
OpenFOAM implementation uses slightly different notation for some coefficients:
Cmuis used for \(\beta^*\)betais used for \(\beta\)gammais used for \(\alpha_\omega\)alphaKis used for \(\sigma_k\)alphaOmegais used for \(\sigma_\omega\)alphaDis used for \(\sigma_d\)
Moreover, OpenFOAM works with kinematic turbulent viscosity nut multiplied by density instead of turbulent viscosity mu_t.