## 2 Numerical Simulations on Jet Impingement
### 3D Model
In Figure 9a,b, the trends of the predicted Nusselt number distributions along the wall for x/D = 0 and x/D = 4 are respectively shown. They correspond to the planes passing through the axis centers of the jets. The above data are compared with the results from [PERSON] [8]. The realizable model k-\(\epsilon\) provides a linear distribution of heat exchange compared to the other models, showing a better match with the experimental data trend. All the turbulence closures considered give an overestimation of the  
Figure 8: Nusselt contours for different turbulence models. Top left: experimental data by [PERSON]. Top right: contours with realizable k-\(\epsilon\) model. Bottom left: contours with k-\(\omega\) SST model. Bottom right: contours with v2f model. The variable x/D (0-6) in the abscissa, y/D (0-6) in the ordinates. peak value. The k-\(\epsilon\) is able to give a closer match with the reference shape with a lower overestimation of the Nusselt number compared to the other models. The peak value is largely overestimated (about 25%) but it occurs in a very small area close to the jet center with negligible effects on the overall heat exchange. In Figure 10, the velocity contours at plane x/D = 0 from the numerical simulation (above) and the experimental data (below) are compared. A difference is observed at the jet axis. Compared to the experimental case, the numerical data have a lower speed decrease on the jet axis. This may further account for the overestimation of the heat exchange in the stagnation area. In Table 1, a comparison of various results is reported, based on the mean convective heat transfer coefficient on the wall. Such a parameter is selected because some correlations are based on a different reference temperature, i.e., the film temperature T\({}_{film}\), defined as follows:  
\[T_{film}=\frac{T_{inlet}+T_{wall}}{2} \tag{4}\]  
It is incorrect to compare Nusselt values calculated with respect to different temperatures; it is preferable to compare the heat transfer coefficient that is a physical quantity independent of the reference temperature selection. The correlations of [PERSON] and [PERSON] [6] and [PERSON] and [PERSON] [6] were applied on the basis of both temperatures, since the available literature did not specify what the reference temperature was. The most accurate correlations are those proposed by [PERSON] and by [PERSON]. The error of the CFD results is about 10% with respect to an average value. This percentage is acceptable from an industrial design point of view. With the above positive results, the proposed CFD model for jet clusters can be considered a valid tool for the simulation of the industrial jet cooler module made of several jet clusters. Figure 9: Experimental and numerical trends of the Nusselt number: (**a**) x/D = 0 and (**b**) x/D = 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Nu (Average)** &
\begin{tabular}{c} **Convective Heat Transfer** \\ **Coefficient h [W/m\({}^{2}\) K]** \\ \end{tabular} & **Percentage Error** \\ \hline Geers—exp & 83.13 & 165.18 & — \\ \hline CFD & 92.20 & 183.19 & 10.90 \\ \hline Correlations & & & \\ \hline Geers (inlet) & 81.20 & 161.39 & \(-\)2.29 \\ \hline Milani (film) & 289.58 & 158.88 & \(-\)3.82 \\ \hline Floreschutz (film) & 97.03 & 192.79 & 16.70 \\ \hline Huber-Viskanta (inlet) & 90.01 & 178.84 & 8.27 \\ \hline Huber-Viskanta (film) & 73.38 & 145.80 & \(-\)11.73 \\ \hline Goldstein-Seol (inlet) & 51.56 & 102.45 & \(-\)37.97 \\ \hline Goldstein-Seol (film) & 42.16 & 83.77 & \(-\)49.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of CFD, experimental, and numerical correlation [6] data. Figure 10: Velocity contours for the plane x/D = 0 (numerical above, experimental below).