## 3 Numerical Results
### Cylindrical Pipe Tests  
With the reference to a duct geometry that consists of a cylinder with a diameter \(D=0.0605\) m, we simulate several cases of fully developed turbulent flows with both \(Pr=0.01\) and \(Pr=0.025\). A summary of these cases is reported in Table 6 together with the relative value of the turbulent Reynolds number \(Re_{\tau}\) and of the Reynolds number \(Re\). On the wall surface, a uniform heat flux \(q\) per unit area is applied with value of \(3.6\times 10^{5}\) W/m\({}^{2}\). As for the case of plane channel, we denote with \(\Gamma_{w}\) the wall surface; with \(\Gamma_{in}\) and \(\Gamma_{out}\) the inlet and outlet sections; and with \(\Gamma_{sym}\) the cylinder axis which is the axial symmetry axis of the problem. The boundary conditions are reported in Table 7, where \(r\) is the radial coordinate, \(u\) its component of the mean velocity field and \(v\) the axial one. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Case** & **A** & **B** & **C** & **D** & **E** & **F** & **G** \\ \hline \(Re_{\tau}\) & 180 & 360 & 550 & 1000 & 3580 & 5840 & 6860 \\ \(Re\) & 5760 & 12,770 & 20,700 & 41,000 & 165,000 & 286,000 & 341,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Cylindrical pipe tests. Summary table and labels for different simulated cases. Figure 2: Two-dimensional channel tests (\(Pr=0.01\)). Non-dimensional mean temperature \(\hat{T}^{+}\) profiles (**a**) and non-dimensional turbulent heat flux \(q_{R}^{+}\) and total heat flux \(q_{eff}^{+}\) (**b**) as a function of the non-dimensional distance from the wall \(y^{+}\). The results obtained with KL\(W\) model are compared with DNS data [10]. On the bottom, quantitative comparisons with DNS data for \(\hat{T}^{+}\) and \(q_{eff}^{+}\). The physical properties are shown in Table 4 where we consider two fluids with different Prandtl numbers. We implement the modified KLW model and also the Kays correlation in the same numerical code. The same test cases are carried out for this model and the same comparison with the experimental correlations is given. In this way we intend to compare two different thermal turbulence models using the same numerical code and the same model for the dynamical turbulence, thereby allowing us to highlight the differences obtained in the simulated heat transfer performances, which are then due only to the different thermal turbulence models. In Figure 3a the non-dimensional temperature \(\hat{T}^{+}\) profiles are reported for the simulated cases. We see that the simulations match the linear behavior \(\hat{T}^{+}=Pr\,y^{+}\), in accordance with the near wall behavior described in Table 2. In Figure 3b the non-dimensional root-mean-squared temperature fluctuation \(k_{\theta}\) is shown as a function of non-dimensional wall distance \(y^{+}\) for all the simulated cases and for both \(Pr=0.01\) and \(0.025\). We can see how the different molecular Prandtl numbers affect the temperature fluctuations. In fact, for any given Reynolds number, the simulations with \(Pr=0.01\) are characterized by a maximum value of \(T_{rms}^{+}\) which is smaller than the one obtained with \(Pr=0.025\) and the peak position is shifted nearer the pipe's middle axis. In Table 8 we report the mean values across the pipe transverse section of non-dimensional mean temperature and non-dimensional RMS temperature fluctuations  
\[\hat{T}^{+}_{m}=\frac{2}{D}\int\limits_{0}^{D/2}\hat{T}^{+}dr\qquad T_{rms,m}^ {+}=\frac{2}{D}\int\limits_{0}^{D/2}T_{rms}^{+}dr\,, \tag{35}\]  
together with their ratio \(T_{rms,m}^{+}/\hat{T}^{+}_{m}\). These parameters allow us to compare the mean temperature fluctuations modulus with the average difference between the fluid and the solid temperatures. For low velocities, i.e., with \(Re_{\tau}\leq 1000\), we see that the temperature fluctuations increase for increasing mean velocities and are higher for \(Pr=0.025\) than for \(Pr=0.01\).