## 3 Numerical Results
### Two-Dimensional Channel Tests  
Experimental data with forced flows in engineering applications cannot provide velocity and temperature profiles due to the measurement limits of the liquid-metal instrumentation. However, these profiles can be computed by direct numerical simulations which are available mainly in low Prandtl numbers, in plane geometries and in fully developed turbulence regime [9; 10]. For the case \(Pr=0.025\), even if the KWL model is slightly modified, the results are very similar to those reported in [8] and therefore we focus our attention to \(Pr=0.01\). Since DNS data in [10] are available for both the dynamical and thermal turbulence for a fluid with \(Pr=0.01\), we study three KLW model simulation cases for \(Re_{\tau}=180\) (A), 395 (B) and 590 (C). The corresponding bulk Reynolds numbers for these cases are respectively \(Re=5760\), 14,160 and 22,170. The physical domain consists of a channel of width \(L=0.0605\) m. On the wall surfaces a uniform heat flux \(q\) per unit area is applied with \(q=3.6\times 10^{5}\) W/m\({}^{2}\). We consider half physical domain because the fully developed turbulent flows present symmetry with respect to the middle channel axis. The boundary conditions we apply are reported in Table 5. We denote with \(x\) the direction normal to the wall surfaces and with \(y\) the axial direction. The inlet and outlet surfaces are represented with \(\Gamma_{in}\) and \(\Gamma_{out}\) respectively; \(\Gamma_{sym}\) indicates the symmetry plane and \(\Gamma_{w}\) the wall surface. Different mesh refinements have been used to study the turbulent flows for the various \(Re_{t}\) cases, in order to obtain a value of non-dimensional wall distance \(y^{+}\) smaller than 1 on the first mesh nodes away from the wall. \begin{table}
\begin{tabular}{c c c c} \hline
**Properties** & & **Value** & \\ \hline Viscosity & \(\mu\) & & 0.00184 & Pa s \\ \hline Density & \(\rho\) & & 10340 & kg/m\({}^{3}\) \\ \hline Thermal conductivity & \(\lambda\) & \begin{tabular}{c} \(Pr=0.01\) \\ \(Pr=0.025\) \\ \end{tabular} & \
\begin{tabular}{c} 26.818 \\ 10.76896 \\ \end{tabular} &
\begin{tabular}{c} \(\text{W/(m K)}\) \\ \end{tabular} \\ \hline Heat specific capacity & \(C_{p}\) & & 146 & J/(kg K) \\ \hline \end{tabular}
\end{table} 
Table 4: Physical properties used in the simulation for \(Pr=0.025\) and 0.01. The results are set to non-dimensional values by using the kinematic viscosity \(\
u\); the friction velocity \(v_{\tau}=\sqrt{\tau_{w}/\rho}\), with \(\tau_{w}\) the wall shear stress and \(\rho\) the fluid density; and the friction temperature \(T_{\tau}=q/(v_{\tau}\rho C_{p})\), where \(C_{p}\) is the fluid specific heat capacity. In Figure 1 the non-dimensional axial velocity profiles \(v^{+}\) are reported for different \(Re_{t}\) together with the DNS results. We see that the linear behavior \(v^{+}=y^{+}\) is well reproduced in the viscous region, while the typical logarithmic behavior \(v^{+}=1/\kappa\ln(y^{+})+5.2\) is better captured by the case of \(Re_{t}=590\), since for the other two cases the \(Re_{t}\) is quite small and the deflection of the velocity profile, close to the channel center, occurs before the complete development of the logarithmic profile. The non-dimensional Reynolds stresses \(\tau_{R}^{+}=\langle u^{\prime+}v^{\prime+}\rangle\) and the non-dimensional total stresses \(\tau_{eff}^{+}=\tau_{R}^{+}+\
u v_{y}/u_{\tau}^{2}\) are shown as a function of the non-dimensional wall distance \(y^{+}\) in Figure 1b together with the DNS data. Figure 1: Two-dimensional channel tests (\(Pr=0.01\)). Non-dimensional mean axial velocity \(v^{+}\) profiles (**a**) and non-dimensional Reynolds stresses (**b**) as a function of the non-dimensional wall distance \(y^{+}\). Results obtained with KLW model are compared with DNS data [10]. The quantitative analysis (plots on bottom) show the percentage difference computed between the KLW and DNS results for \(v^{+}\) and \(\tau_{eff}^{+}\).