Wilcox k-w

The \(k\)- \(\omega\) two-equation model is based on a formulation by Wilcox [64] . It has been continuously developed, extended and validated since its inception. In TRACE, a number of extensions are activated by default. Below is a description of the model itself and the most important extensions available in TRACE

Original model

The Reynolds stress tensor for the \(k\)- \(\omega\) model is computed according to

\begin{eqnarray} \tau_{ij}^* = 2 \mu_T^* \left( S_{ij}^* -\frac{1}{3}\frac{\partial u_k^*}{\partial x_k^*} \right) - \frac{2}{3} \rho^* k^* \delta_{ij} \end{eqnarray}

where the turbulent kinetic energy \(k\) is defined as

\begin{eqnarray} k = \frac{1}{2} \overline{u_i'u_i'} \end{eqnarray}

The mean strain rate and vorticity tensors are defined as, respectively,

\begin{eqnarray} S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \end{eqnarray}

and

\begin{eqnarray} \Omega_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \end{eqnarray}

To determine the unknowns, two transport equations of the turbulent kinetic energy \(k\) and the specific dissipation rate \(\omega\) are introduced and have to be solved:

\begin{eqnarray} \frac{\partial}{\partial t} \left( \rho k \right) + \frac{\partial}{\partial x_j} \left( \rho u_j k \right) = \tau_{ij} \frac{\partial u_i}{\partial x_j} - \beta_k \rho \omega k + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_k \mu_T \right) \frac{\partial k}{\partial x_j} \right] \end{eqnarray}

\begin{eqnarray} \frac{\partial}{\partial t} \left( \rho \omega \right) + \frac{\partial}{\partial x_j} \left( \rho u_j \omega \right) = \alpha \frac{\omega}{k} \tau_{ij} \frac{\partial u_i}{\partial x_j} - \beta_\omega \rho \omega^2 + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_\omega \mu_T \right) \frac{\partial \omega}{\partial x_j} \right] \end{eqnarray}

Once \(k\) and \(\omega\) are determined the following relation leads to the turbulent viscosity \(\mu_T\):

\begin{eqnarray} \mu_T = \frac {\rho k}{\omega} \Leftrightarrow \nu_T = \frac{k}{\omega} \end{eqnarray}

The constants take the values

\(\alpha\)	\(\beta_\omega\)	\(\beta_k\)	\(\sigma_k\)	\(\sigma_\omega\)
5/9	3/40	9/100	1/2	1/2

Modifications available for the Wilcox 1988 model

• Stagnation point anomaly:
  • Kato-Launder-Modification
  • Realizability constraint based on the Cauchy-Schwarz inequality
• Compressibility effects:
  • Correction for the compressible mixing layer (dilatation dissipation)
  • Pressure dilatation
• Rotational effects:
  • Effects of rotation on the decay of turbulence
  • Effects of streamline curvature on amplification and damping of turbulence
  • Vortex extension

Model Equations with all default extensions activated

With all the modifications that are by default included, the turbulence model equations can be rewritten as

\begin{eqnarray} \frac{D \rho k}{Dt} = \frac{\partial}{\partial x_j} \left[ (\mu + \sigma_k \mu_t) \frac{\partial k}{\partial x_j} \right] + P_k - D_k + \underbrace{\alpha_2 M_t P_k + \alpha_3 M_t^2 D_k}_{\text{pressure dilatation}} - \underbrace{\frac{3}{2} \max \left( M_t^2 - \frac{1}{16}, 0 \right) D_k }_{\text{turbulent mixing layer}} \end{eqnarray}

\begin{eqnarray} \frac{D \rho \omega}{Dt} &=& \frac{\partial}{\partial x_j} \left[ (\mu + \sigma_k \mu_t) \frac{\partial \omega}{\partial x_j} \right] + P_{\omega} - D_{\omega} \\ &&- \underbrace{\alpha_2 M_t \frac{\omega}{k} P_k + \alpha_3 M_t^2 \frac{\omega}{k} D_k}_{\text{pressure dilatation}} - \underbrace{\frac{3}{2} \frac{\beta_k}{\beta_{\omega}} \max \left( M_t^2 - \frac 1 {16}, 0 \right) D_{\omega}}_{\text{turbulent mixing layer}} - \underbrace{c_2 \rho \omega \Omega_\mathrm{norm}}_{\text{rotation}} \end{eqnarray}