## 3 Numerical Results
### Two-Dimensional Channel Tests
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Var.** & \(\mathbf{\Gamma}_{sym}\) & \(\mathbf{\Gamma}_{in}\cup\mathbf{\Gamma}_{out}\) & \(\mathbf{\Gamma}_{w}\) \\ \hline \(u\) & \(u=0\) & \(\frac{3u}{3\pi}=0\) & \(u=0\) \\ \(v\) & \(\frac{3u}{3\pi}=0\) & \(\frac{3u}{3\pi}=0\) & \(\frac{3u}{3\pi}=\frac{3u}{3\pi}\) \\ \(k\) & \(\frac{3u}{3\pi}=0\) & \(\frac{3u}{3\pi}=0\) & \(\frac{3u}{3\pi}=\frac{3u}{3\pi}\) \\ \(\Omega\) & \(\frac{\Omega}{\alpha\Omega}=0\) & \(\frac{\Omega}{\alpha\Omega}=0\) & \(\Omega=\ln\left(\frac{2v}{\zeta_{\
u}\rho^{\ast}}\right)\) \\ \(T\) & \(\frac{3\pi}{3\pi}=0\) & \(\frac{3\pi}{3\pi}=0\) & \(T=0\) \\ \(k_{\theta}\) & \(\frac{3u}{3\pi}=0\) & \(\frac{3u}{3\pi}=0\) & \(\frac{3u}{3\pi}=\frac{2u}{3\pi}\) \\ \(\Omega_{\theta}\) & \(\frac{\partial\Omega_{\theta}}{\alpha\mu}=0\) & \(\frac{\partial\Omega_{\theta}}{\partial y}=0\) & \(\Omega_{\theta}=\ln\left(\frac{2u}{\zeta_{\
u}\rho^{\ast}}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Two-dimensional channel tests. Boundary conditions. The comparison is good in the logarithmic region (\(y^{+}>30\)), while some discrepancies are present in the buffer region (\(5<y^{+}<30\)) and in a part of the viscous region. A more quantitative comparison between the KLW results and the DNS ones is reported on the bottom plots of Figure 1, where the percentage difference  
\[{}_{\rm DIF}=\frac{\lambda_{\rm KLW}-\lambda_{\rm DNS}}{\lambda_{eff,\rm DNS}} \tag{34}\]  
is defined on the base of the effective value. For the non-dimensional velocity, the difference with respect to the DNS results is bounded in the range \([-2\%;6\%]\), with slightly higher values obtained in the buffer region. Even if the non-dimensional Reynolds stress component \(\tau_{r}^{+}\) shows some underestimation with respect to the DNS value in the viscous region, the total stress \(\tau_{eff}^{+}=\tau^{+}+\tau_{R}^{+}\) shows very good agreement with the reference values due to the the turbulent stress. In fact, the viscous stress is predominant in the viscous region, while the turbulent stress becomes significant in the buffer and bulk regions. For the total stress the difference with DNS data is in the range \([-0.5\%,2\%]\). The non-dimensional mean temperature profiles \(\tilde{T}^{+}\) are shown in Figure 2a for all the simulated cases. We see that the numerical results match the linear behavior \(\tilde{T}^{+}=Pr\,y^{+}\) and the relative DNS data. In Figure 2b we report the profiles of the non-dimensional turbulent heat flux \(\mathbf{q}_{R}^{+}\cdot\hat{n}=\langle u^{\prime+}T^{\prime+}\rangle\), where \(\hat{n}\) is the unit vector normal to the wall surface, and of the total non-dimensional heat flux \(q_{eff}^{+}=(\mathbf{q}_{R}^{+}+\mathbf{q}^{+})\cdot\hat{n}\). In the viscous region the non-dimensional heat flux profiles agree with the reference behavior, i.e.,  
\[\langle u^{\prime+}T^{\prime+}\rangle\propto(y^{+})^{3}\,,\]  
which is obtained in the case of zero temperature fluctuations on the heated wall. In fact, from the near wall behaviors described in Table 2, it follows  
\[\langle u^{\prime+}T^{\prime+}\rangle=(b_{2}y^{2}+\dots)\cdot(d_{0}+d_{1}y+ \dots)\,.\]  
Having imposed null temperature fluctuations on the heated wall, the value of \(d_{0}\) is equal to zero, so \(\langle u^{\prime+}T^{\prime+}\rangle=b_{2}d_{1}y^{3}+\dots\), with \(b_{2}\) and \(d_{1}\) being unknown coefficients. As for the case of non-dimensional velocity and Reynolds stress component, a quantitative analysis is reported with the percentage difference plots shown on the bottom of Figure 2. For the non-dimensional temperature, the agreement is very satisfactory, with a diff value in the range \([-0.25\%;0.5\%]\), while for the total non-dimensional heat flux the difference with DNS is in the range \([-0.5\%,\,2.5\%]\). In particular, in the viscous and buffer regions the computed turbulent heat flux overestimates the reference values.