## 2 Numerical Simulations on Jet Impingement
### 3D Model  
To check the effects of the interaction among different jets, a 3D case is analyzed: it is characterized by nine in-line jets arranged in three rows. [PERSON] [8] experimentally investigated the same geometry considered; in Figure 5, the top view of the geometry is shown. A perforated plate generates the jet cluster; the diameter D = 13 mm, the distance s between hole centers is 52 mm, and the jet-wall distance is H = 52 mm. The dimensionless ratios H/D = 4 and s/D = 4 are obtained. Figure 4: Nusselt number along the wall H/D = 6—CFD vs. experimental data. Figure 3: Nusselt number along the wall H/D = 2—CFD vs. experimental data. Taking advantage of the geometrical symmetry, only a quarter of the actual geometry is modeled. The symmetry condition is imposed on the outer surfaces of the volume passing through the axes of the modeled jets. The inlet boundary conditions for jet velocity and turbulence profiles are obtained from a separate model that simulates the flow from the orifice. In Figure 6, an overview of the domain with the block-structured mesh is shown, and in Figure 7, the detail of the orifice mesh for the inlet velocity and turbulence profile is reported. After a sensitivity analysis on the grid, to find the proper refinement, a mesh with about \(1.1\times 10^{6}\) elements is selected to give a value of \(\mathrm{y}^{+}<1\) at the wall. Three simulations are carried out, with the  
Figure 5: Geometry of the experimental analysis by [PERSON]. (**a**) Top view and (**b**) side view. Figure 6: Geometric model for 3D case. Figure 7: Detail of the geometry adopted to simulate the flow coming out of an orifice. same Reynolds number (Re = 20,000). The kinematic viscosity for the above Reynolds number is evaluated at the temperature of the jet. Different turbulence closures are compared for the 3D flow structure analysis: realizable k-\(\epsilon\) (with the use of enhanced wall treatment), v2f, and k-\(\omega\) SST. The jet reference temperature was chosen to compare the results with [PERSON] data; the thermal conductivity was also evaluated at the same temperature as in the Nusselt number calculation. A constant heat flux is set at the wall, and the pressure is set equal to the environment on the outlet surfaces. As in the previous case, the SIMPLE spatial scheme is used and the variability of the thermo-physical properties with temperature is considered. The first-order upwind discretization scheme is preferred over the higher order scheme because of its higher stability. Numerical instabilities arise with the higher order upwind scheme. In Figure 8, the numerical results from the different turbulence closures are compared to the reference data. After a qualitative comparison based on the experimental and numerical contours, it can be noted that the realizable k-\(\epsilon\) model provides the best results. In contrast to [PERSON] et al. [9] and consistent with [PERSON] al. [14] and [PERSON] et al. [12], for a 2D model, the realizable k-\(\epsilon\) model can provide comparable or better results than the v2f option. Such a difference, if compared to the 2D case, can be attributed to a higher robustness of the k-\(\epsilon\) model, which is stronger and more stable, in obtaining the convergence and the wall temperature stabilization. Furthermore, the incoming turbulence is very different between a fully developed flow and the flow generated by an orifice. In the former, it is greater at the center of the jet, while in the latter, the turbulence is higher along the edges of the jet, because of the vena contracta formation. These different velocity and turbulent profiles give different jets from a physical point of view. The turbulent profile from the orifice has lower values of turbulent kinetic energy on the jet axis. From a numerical point of view, this lower value of incoming kinetic energy gives a lower overestimation of k, typical of the k-\(\epsilon\) model in stagnation zones.