## 3 Numerical Results
### Cylindrical Pipe Tests
For the cases \(Re_{\tau}>1000\) we observe that the importance of the temperature fluctuations, relative to the mean temperature, decreases for increasing velocities and that it is higher for \(Pr=0.01\) than for \(Pr=0.025\). \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Var.** & \(\mathbf{\Gamma}_{sym}\) & \(\mathbf{\Gamma}_{in}\cup\mathbf{\Gamma}_{out}\) & \(\mathbf{\Gamma}_{w}\) \\ \hline \(u\) & \(u=0\) & \(\frac{\partial u}{\partial y}=0\) & \(u=0\) \\ \(v\) & \(\frac{\partial v}{\partial x}=0\) & \(\frac{\partial v}{\partial y}=0\) & \(\frac{\partial v}{\partial x}=\frac{\mu}{\partial}\) \\ \(k\) & \(\frac{\partial k}{\partial x}=0\) & \(\frac{\partial k}{\partial y}=0\) & \(\frac{\partial k}{\partial x}=\frac{2k}{\partial}\) \\ \(\Omega\) & \(\frac{\partial\Omega}{\partial y}=0\) & \(\frac{\partial\Omega}{\partial y}=0\) & \(\Omega=\ln\left(\frac{2v}{c_{\tau}a^{2}}\right)\) \\ \(\hat{T}\) & \(\frac{\partial T}{\partial x}=0\) & \(\frac{\partial T}{\partial y}=0\) & \(\hat{T}=0\) \\ \(k_{\theta}\) & \(\frac{\partial k\theta}{\partial x}=0\) & \(\frac{\partial k\theta}{\partial y}=0\) & \(\frac{\partial k\theta}{\partial x}=\frac{2k\theta}{\partial x}\) \\ \(\Omega_{\theta}\) & \(\frac{\partial\Omega}{\partial x}=0\) & \(\frac{\partial\Omega x}{\partial y}=0\) & \(\Omega_{\theta}=\ln\left(\frac{2k\theta}{c_{\tau}a^{2}}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Cylindrical pipe tests. Boundary conditions. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(Re_{\tau}\)} & \multicolumn{4}{c}{\(Pr=0.01\)} & \multicolumn{4}{c}{\(Pr=0.025\)} \\ \cline{3-6}  & & \(\hat{T}^{+}_{m}\) & \(T^{\prime+}_{rms,m}\) & \(T^{\prime+}_{rms,m}/\hat{T}^{+}_{m}\) & \(\hat{T}^{+}_{m}\) & \(T^{\prime+}_{rms,m}\) & \(T^{\prime+}_{rms,m}/\hat{T}^{+}_{m}\) \\ \hline A & 180 & 0.62 & 0.025 & 0.041 & 1.46 & 0.112 & 0.076 \\ B & 360 & 1.17 & 0.075 & 0.064 & 2.53 & 0.275 & 0.108 \\ C & 550 & 1.71 & 0.134 & 0.078 & 3.44 & 0.402 & 0.117 \\ D & 1000 & 2.82 & 0.268 & 0.095 & 5.16 & 0.609 & 0.117 \\ E & 3578 & 6.35 & 0.711 & 0.111 & 9.30 & 0.895 & 0.096 \\ F &For forced convection flows and high Reynolds numbers DNS simulations are not available and only experimental data can be compared with the results obtained with the KLW model. As a consequence, temperature profiles actually cannot be captured from liquid-metal experiments, and only integral measurements, i.e., Nusselt number values and heat transfer coefficients, are available. Nusselt numbers usually define the heat transfer coefficients between a wall surface and a fluid flow as  
\[Nu=\frac{hD_{h}}{\lambda}\,, \tag{36}\]  
where \(\lambda\) is the thermal conductivity, \(D_{h}\) the hydraulic diameter and \(h\) the convective heat transfer coefficient. If a constant heat flux \(q\) is applied on wall surfaces, the convective heat transfer coefficient \(h\) can be written as  
\[h=\frac{q}{T_{w}-T_{b}}\,, \tag{37}\]  
where \(T_{w}\) is the wall surface temperature and \(T_{b}\) is the bulk temperature of the fluid defined as  
\[T_{b}=\frac{\int_{A}T\mathbf{u}\cdot\hat{n}dA}{\int_{A}\mathbf{u}\cdot\hat{n} dA}\,, \tag{38}\]  
where the unit vector normal is denoted by \(\hat{n}\) and \(A\) is the channel cross section. Experimental data summarize their results in the form of correlations for each type of geometry as a function of the Peclet number \(Pe=Pr\cdot Re\). Many experimental correlations are known for liquid metals, but unfortunately many of them do not agree. One can easily see that there is a disagreement between the different experiments and authors have attempted many explanations for this discrepancy; see for example, [4]. The general expressions of these correlations for the cylindrical pipe geometry take the following form:  
\[Nu=A+a\,Pe^{n}\,, \tag{39}\]  
where \(a\) and \(n\) are constant assumed to be positive numbers.