## 2 Geometry and CFD-Thermal Modeling
### Convergence under Mesh Refinement Study  
A grid independence study is conducted for the highest jet \(Re\) number considered in this study. \(D_{p}\) and \(H_{p}\) are selected as 3.5\(D\) and 1\(D\), respectively. For 3D CFD simulations, a hybrid grid consisting of triangular base prism layers and tetrahedrons was created due to  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Description** & **Value** \\ \hline \(D\) & Jet diameter at the orifice exit & 4 mm \\ \(Re\) & Jet Reynolds number at the orifice exit & 5000, 10,000, 15,000, 20,000 \\ \(H_{p}/D\) & Dimensionless protrusion height & 1, 15, 2, 2.5, 3, 4, 4.5 \\ \(D_{p}/D\) & Dimensionless protrusion diameter & 2, 2.5, 3, 3.5, 4 \\ \(\overline{P}_{r}\) & Prandtl number of air & 0.71 \\ \(\overline{T}_{I}\) & Jet temperature at the orifice exit & 293 K \\ \(\overline{T}_{w}\) & Target plate temperature & 303 K \\ \(\rho\) & Air density & 1.204 kg/m\({}^{3}\) \\ \(\overline{u}_{I}\) & Jet velocity at the orifice exit & \(\overline{u}_{I}=(Re\mu)/(\rho D)\) \\ \(I\) & Jet turbulence intensity at the orifice exit & \(I=0.16~{}Re^{(-1/8)}\) \\ \(\mu\) & Dynamic viscosity of air & 1.813 \(\times\) 10\({}^{-5}\) kg/ms \\ \(\dot{Q}\) & Overall heat transfer rate (W) & 1.813 \(\times\) 10\({}^{-5}\) kg/ms \\ \(\dot{Q}_{o}\) & Heat transfer rate (W) & \\ \(\dot{Q}/\dot{Q}_{o}\) & Heat transfer ratio & \\ \(\overline{Nu}_{o}\) & Area-averaged Nusselt number & \\ \(\overline{Nu}_{o}\) & Area-averaged Nusselt number of the flat target plate & \\ \(\overline{Nu}/\overline{Nu}_{o}\) & Ratio of the area-averaged Nusselt numbers & \\ \(PEC\) & Performance evaluation criterion & \\ \hline \end{tabular}
\end{table}
Table 1: The geometric and operating parameters of CMJICS and the thermo-physical properties of air.  
Figure 5: Solution domain and grid: (**a**) solution domain and boundary conditions; (**b**) grid with triangular prism layers near walls and tetrahedra in the bulk of the solution domain.  
the complex geometry of the CMJICS problem. Twenty layers of prisms with triangular bases (wedge-shaped finite volumes) adjacent to the walls and a gradually expanding tetrahedral mesh throughout the remainder of the solution domain are created. The total height of the prism layers and the height ratio between layers, which is kept below 1.2 for smoothness, were adjusted to accurately resolve the momentum and thermal boundary layers on the walls. Therefore, the nonuniform hybrid meshes created in this study cluster finite volumes in regions adjacent to walls. Prism layers cover the boundary layers on both the cooled surfaces and the perforated confinement plate. Since the cell size varies across the wall surfaces, the maximum aspect ratio of the first prism layer cells is limited to less than 55, thus preventing them from becoming too large, especially toward the outlet. To accurately resolve turbulent momentum and thermal boundary layers, the maximum \(y+\) value of the centroids of cells adjacent to walls is kept close to unity on cooled surfaces. Moreover, refined tetrahedral cells are generated in the immersed jet regions, above the jet stagnation zones, around the CPs and CGVs as seen in Figure 5b. The tetrahedral mesh is expanded toward the core region and the exit with a maximum growth rate of 1.2. Four meshes of various resolutions named M1, M2, M3, and M4 are created, consisting of 2.66 \(\times\) 10\({}^{5}\), 5.44 \(\times\) 10\({}^{5}\), 1.68 \(\times\) 10\({}^{6}\) and 2.59 \(\times\) 10\({}^{6}\) cells, respectively. For the M1, M2, M3, and M4 meshes, on the cooled surfaces, the calculated highest \(y+\) values are 1.80, 1.68, 1.65, and 1.29, and the \(\overline{Nu}\) numbers are 50.94, 51.58, 52.16 and 52.32, respectively. The simulation results obtained on M3 are presented in the following section due to only a 0.3% difference between the \(\overline{Nu}\) numbers calculated on M3 and M4.