k-log(w)

The \(k\)- \(log(\omega)\) model is a modification of the Wilcox k-w model and can be activated by the -logW switch in the control file. It was implemented to remove the overshoots of the conventional \(k\)- \(\omega\) model when used in the harmonic balance solver. Those occur due to high gradients of \(\omega\), which approaches infinity close to a wall. By replacing \(\omega\) with \(log(\omega)\) those gradients can be decreased.

Derivation of the equations

To obtain the \(k\)- \(log(\omega)\) equations a new variable \(\tilde{\omega}\) is introduced:

\begin{eqnarray} \tilde{\omega}=log(\omega) \Longleftrightarrow \omega=e^{\tilde{\omega}} \end{eqnarray}

The derivatives are also needed for a transformation of the equations:

\begin{eqnarray} \parfrac{\omega}{t}=\parfrac{\omega}{\tilde{\omega}} \parfrac{\tilde{\omega}}{t} = e^{\tilde{\omega}} \parfrac{\tilde{\omega}}{t} \end{eqnarray}

Since both the arguments for \(log()\) and \(e^{()}\) have to be dimensionless, \(\omega\) and \(\tilde{\omega}\) need to be non-dimensionalised:

\begin{eqnarray} \frac{\omega^*}{\omega_0^*}=\omega=e^{\tilde{\omega}} \quad \quad \omega_0^*=\frac{{\rho_0}^* {a_0}^{*2}}{{\mu_0}^*} \end{eqnarray}

Variables with \(*\) denote a quantity with dimensions.

Starting from the Wilcox k-w equations the \(k\) and \(\tilde{\omega}\) can be derived:

\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^* k^* \omega_0^* e^{\tilde{\omega}} + \frac{\partial}{\partial x_j^*} \left[ \left( \mu^* + \sigma_k \mu_T^* \right) \frac{\partial k^*}{\partial x_j^*} \right] \end{eqnarray}

\begin{eqnarray} \parfrac{\rho^*}{t^*} + \rho^*\parfrac{\tilde{\omega}}{t^*} + \parfrac{\rho^* u_j^*}{x_j^*} +\rho^* u_j^* \parfrac{\tilde{\omega}}{x_j} &=& \frac{\alpha }{k^*} \tau_{ij}^* \frac{\partial u_i^*}{\partial x_j^*} - \beta_\omega \rho^* \omega_0^* e^{\tilde{\omega}} + \frac{\partial}{\partial x_j^*} \left[ \left( \mu^* + \sigma_\omega \mu_T^* \right) \frac{\partial \tilde{\omega}}{\partial x_j^*} \right] \\ &&+ \left( \mu^* + \sigma_\omega \mu_T^* \right) \frac{1}{e^{\tilde{\omega}}} \parfrac{e^{\tilde{\omega}}}{x_j^*} \parfrac{\tilde{\omega}}{x_j^*} \end{eqnarray}

With the continuity equation applied to the left site and a expansion of the last term on the right site

\begin{eqnarray} \parfrac{\rho^*}{t^*} + \parfrac{\rho^* u_j^*}{x_j^*}=0 \Longleftrightarrow \tilde{\omega} \left( \parfrac{\rho^*}{t^*} + \parfrac{\rho^* u_j^*}{x_j^*} \right)=0 \quad \quad \frac{1}{e^{\tilde{\omega}}} \parfrac{e^{\tilde{\omega}}}{\tilde{\omega}} \parfrac{\tilde{\omega}}{x_j}=\frac{1}{e^{\tilde{\omega}}} \parfrac{e^{\tilde{\omega}}}{\tilde{\omega}} \parfrac{\tilde{\omega}}{x_j} \parfrac{\tilde{\omega}}{x_j} = \parfrac{\tilde{\omega}}{x_j} \parfrac{\tilde{\omega}}{x_j} \end{eqnarray}

the \(\tilde{\omega}\) equation can be simplified and brought into conservative form:

\begin{eqnarray} \parfrac{\rho^*\tilde{\omega}}{t^*} + \parfrac{\rho^* u_j^*\tilde{\omega}}{x_j^*} = \frac{\alpha }{k^*} \tau_{ij}^* \frac{\partial u_i^*}{\partial x_j^*} - \beta_\omega \rho^* \omega_0^* e^{\tilde{\omega}} + \frac{\partial}{\partial x_j^*} \left[ \left( \mu^* + \sigma_\omega \mu_T^* \right) \frac{\partial \tilde{\omega}}{\partial x_j^*} \right] + \left( \mu^* + \sigma_\omega \mu_T^* \right) \parfrac{\tilde{\omega}}{x_j^*} \parfrac{\tilde{\omega}}{x_j^*} \end{eqnarray}

Additionally, the definition of \({\mu}_T^*\) has to be changed:

\begin{eqnarray} \mu_T^* = \frac {\rho^* k^* }{\omega_0^* e^{ \tilde{\omega}}} \end{eqnarray}

The set of transport equations in fully non-dimensionalised form are:

\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 Re \rho k e^{\tilde{\omega}} +\frac{1}{Re} \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_k \mu_T \right) \frac{\partial k}{\partial x_j} \right] \end{eqnarray}

\begin{eqnarray} \parfrac{\rho\tilde{\omega}}{t} + \parfrac{\rho u_j\tilde{\omega}}{x_j} = \frac{\alpha }{k} \tau_{ij} \frac{\partial u_i}{\partial x_j} - Re \beta_\omega \rho e^{\tilde{\omega}} + \frac{1}{Re} \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_\omega \mu_T \right) \frac{\partial \tilde{\omega}}{\partial x_j} \right] + \frac{1}{Re} \left( \mu + \sigma_\omega \mu_T \right) \parfrac{\tilde{\omega}}{x_j} \parfrac{\tilde{\omega}}{x_j} \end{eqnarray}

with

\begin{eqnarray} \tau_{ij}=\frac{2\mu_T}{Re_L} \left( S_{ij} - \frac{1}{3} \parfrac{u_k}{x_j} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij} \quad \quad \end{eqnarray}