## Mathematical Model
### Turbulence Model \(k\)-\(\Omega\)-\(k_{\theta}\)-\(\Omega_{\theta}\)
By substituting the new definitions in Equation (7) in the \(k\)-\(\varepsilon\) formulation of the turbulence model, we obtain [8; 16; 17; 18]  
\[\frac{\partial k}{\partial t}+\mathbf{u}\cdot\boldsymbol{\
abla}k -\boldsymbol{\
abla}\cdot\left[\left(\
u+\frac{\
u_{t}}{\sigma_{k}}\right) \boldsymbol{\
abla}k\right]=\mathcal{P}_{k}-C_{\mu}\,k\,e^{\Omega}\,, \tag{8}\] \[\frac{\partial\Omega}{\partial t}+\mathbf{u}\cdot\boldsymbol{ \
abla}\Omega-\boldsymbol{\
abla}\cdot\left[\left(\
u+\frac{\
u_{t}}{\sigma_{ \varepsilon}}\right)\boldsymbol{\
abla}\Omega\right]=\] (9) \[\frac{2}{k}\left(\
u+\frac{\
u_{t}}{\sigma_{\varepsilon}}\right) \boldsymbol{\
abla}k\cdot\boldsymbol{\
abla}\Omega+\left(\
u+\frac{\
u_{t}}{ \sigma_{\varepsilon}}\right)\boldsymbol{\
abla}\Omega\cdot\boldsymbol{\
abla }\Omega+\frac{c_{\varepsilon 1}-1}{k}\mathcal{P}_{k}-C_{\mu}\left(c_{\varepsilon 2}f_{ \varepsilon}-1\right)e^{\Omega}\,,\] \[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\mathbf{\
abla} \mathbf{u}=-\frac{1}{\rho}\mathbf{\
abla}P+\mathbf{\
abla}\cdot\left[(\
u+\
u_{t}) \left(\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^{T}\right)\right]+\mathbf{g}\,,\]0 (10) \[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\mathbf{\
abla} \mathbf{u}=-\frac{1}{\rho}\mathbf{\
abla}P+\mathbf{\
abla}\cdot\left[(\
u+\
u_{t}) \left(\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^{T}\right)\right]+\mathbf{g}\,,\]1 (11) \[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\mathbf{\
abla} \mathbf{u}=-\frac{1}{\rho}\mathbf{\
abla}P+\mathbf{\
abla}\cdot\left[(\
u+\
u_{t}) \left(\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^{T}\right)\right]+\mathbf{g}\,,\]2  
The terms \(\mathcal{P}_{k}\) and \(\mathcal{P}_{k\theta}\) are the source terms for turbulent kinetic energy and temperature fluctuations. In our modeling we set  
\[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\mathbf{\
abla} \mathbf{u}=-\frac{1}{\rho}\mathbf{\
abla}P+\mathbf{\
abla}\cdot\left[(\
u+\
u_{t}) \left(\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^{T}\right)\right]+\mathbf{g}\,,\]3  
A summary of the model's constant values is collected in Table 1, where \(f_{\varepsilon}\) and \(C_{d2}\) are two functions needed to correct the behavior of the turbulence model near the wall [15; 16; 17; 18]  
\[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\mathbf{\
abla} \mathbf{u}=-\frac{1}{\rho}\mathbf{\
abla}P+\mathbf{\
abla}\cdot\left[(\
u+\
u_{t}) \left(\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^{T}\right)\right]+\mathbf{g}\,,\]4  
\[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\mathbf{\
abla} \mathbf{u}=-\frac{1}{\rho}\mathbf{\
abla}P+\mathbf{\
abla}\cdot\left[(\
u+\
u_{t}) \left(\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^{T}\right)\right]+\mathbf{g}\,,\]5  
We define the turbulent Reynolds number \(R_{t}\) and the non-dimensional length \(R_{\delta}\) by using the variables \(k\) and \(\varepsilon\). Let \(\delta\) be the wall distance and \((\upsilon^{3}/\varepsilon)^{1/4}\) be the Kolmogorov length scale. Now we have  
\[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\mathbf{\
abla} \mathbf{u}=-\frac{1}{\rho}\mathbf{\
abla}P+\mathbf{\
abla}\cdot\left[(\
u+\
u_{t}) \left(\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^{T}\right)\right]+\mathbf{g}\,,\]6  
and, when logarithmic variables are used, \(R_{t}\) and \(R_{\delta}\) become  
\[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\mathbf{\
abla} \mathbf{u}=-\frac{1}{\rho}\mathbf{\
abla}P+\mathbf{\
abla}\cdot\left[(\
u+\
u_{t}) \left(\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^{T}\right)\right]+\mathbf{g}\,,\]7  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(C_{1\varepsilon}\) & \(C_{2\varepsilon}\) & \(C_{\mu}\) & \(\sigma_{k}\) & \(\sigma_{\varepsilon}\) & \(c_{p1}\) & \(c_{p2}\) & \(c_{d1}\) & \(\sigma_{\theta}\) & \(\sigma_{\varepsilon\theta}\) \\ \hline
1.5 & 1.9 & 0.09 & 1.4 & 1.4 & 1.025 & 1.9 & 1.1 & 1.