## 3 Numerical Results
### Cylindrical Pipe Tests
As said before, there are many different experimental correlations and each correlation comes from a different experiment that brings its own  
Figure 3: Cylindrical pipe tests (KLW model). Non-dimensional profiles of the mean temperature \(\hat{T}^{+}\) (**a**) and the root-mean-squared temperature fluctuations \(T_{rms}^{\prime+}\) (**b**) as a function of the non-dimensional distance from the wall \(y^{+}\) with \(Pr=0.025\) and \(0.01\). The cases A, B, C, D, E, F and G are defined in Table 6.  
range of validity expressed in terms of Reynolds numbers. Some of these well known correlations are reported as  
\[ \begin{array}{llll}\text{\it Lyon}&\text{\it Nu}&\text{\it Nu}&\text{\it 7.0 }+0.025\left(\frac{Pe}{Pr_{t}}\right)^{0.8}&\text{\it 10}^{4}&\text{\it Re}& \text{\it e}&\text{\it 5\cdot 10^{6}}\,,\\ \text{\it Skupinski}&\text{\it Nu}&\text{\it Nu}&\text{\it 4.82}+0.0185 Pe^{0.827}& \text{\it 10}^{4}&\text{\it Re}&\text{\it e}&\text{\it 5\cdot 10^{6}}\,,\\ \text{\it [PERSON]}&\text{\it Nu}&\text{\it Nu}&\text{\it 6.3}+0.0167 Pe^{0.85}Pr^{0. 08}&\text{\it 10}^{4}&\text{\it Re}&\text{\it e}&\text{\it 5\cdot 10^{6}}\,,\\ \text{\it [PERSON]}&\text{\it Nu}&\text{\it Nu}&\text{\it 4.5}+0.014 Pe^{0.8}& \text{\it 10}^{4}&\text{\it Re}&\text{\it e}&\text{\it 5\cdot 10^{6}}\,,\\ \text{\it [PERSON]}&\text{\it Nu}&\text{\it Nu}&\text{\it 3.6}+0.018 Pe^{0.8}& \text{\it 88}&\text{\it Pe}&\text{\it 4000}\,.\\ \text{\it [PERSON]}&\text{\it Nu}&\text{\it Nu}&\text{\it 4.5}+0.018 Pe^{0.8}& \text{\it 10}^{4}&\text{\it Re}&\text{\it e}&\text{\it 5\cdot 10^{6}}\,.\\ \end{array}  \tag{40}\]  
The correlations in Equations (40)-(45) can be seen in Figure 4. Each correlation has a name related to its investigator, as labeled in Figure 4. In this Figure, correlation data from different experiments, such as lead, lead-bismuth, bismuth and sodium, are mixed with the aim to obtain a law valid for all liquid metals. The interested reader can see details for each experiment in [29; 30; 31; 32; 33; 34; 35; 36]. For low values of Peclet numbers (\(Pe<1000\)) the Kirillov curve is the correlation that fits better all the experimental points [4; 34]. In [4] the authors proposed the following new correlation  
\[ \begin{array}{ll}\text{\it Nu}=A+0.018 Pe^{0.8}\,,\\ \text{\it A}=\left\{\begin{array}{ll}\text{\it 4.5}&Pe<1000\,,\\ \text{\it 5.4}-9\times 10^{-4}Pe&\text{\it 1000}\leq Pe\leq 2000\,,\\ \text{\it 3.6}&Pe>2000\,.\\ \end{array}\right.\end{array}  \tag{41}\]  
It is easy to see that this correlation matches the Kirillov correlation in the low Peclet region while it tends to the Stromquist one in the range of high Peclet numbers. In Figure 5 we compare the values of \(Nu\) obtained with the KLW and with Kays models for simulations in the range \(80<Pe<10000\) with \(Pr=0.025\) and \(0.01\) with Kirillov and Cheng correlations. The same data are shown in Table 9 as functions of different \(Pr\) and \(Re\) numbers. Figure 4: Cylindrical pipe tests (constant heat flux). Nusselt numbers from different experimental data in Equations (40)–(45) as a function of the Peclet number. The KLW model, for \(Pr=0.025\), matches close to the Kirillov correlation when \(Pe>1000\), while some difference can be seen for \(Pr=0.01\). The KLW model lays between Kirillov and the Cheng correlations, in the high Peclet region, while in the low Peclet region the model overestimates the values. From Figure 5 it can be seen that the Nusselt number values obtained from the KLW model, in both \(Pr=0.01\) and \(Pr=0.025\) cases, lay in the range \([0,+5\%]\) with respect to the Cheng correlation. As it regards the Kays model, we can see that for low Peclet and Prandtl numbers the results are greater than those obtained with KLW model and predicted by both correlations, while with increasing Peclet the mismatch becomes very large, with Nusselt number values lying in the range \([+30\%;+50\%]\) with respect to Cheng correlation.