## 2 Geometry and CFD-Thermal Modeling
### Turbulence Modelling  
The accuracy of jet impingement cooling performance predictions by various RANS turbulence models against experimental data are studied by many researchers [60; 61; 62; 63; 64; 65; 66; 67; 68]. Some studies recommend the shear-stress transport (SST) \(k\omega\) turbulence model for improved accuracy of results. However, in this study, the realizable \(k-\varepsilon\) turbulence model [69] with enhanced wall treatment is preferred because of its numerical stability and faster convergence features compared with the SST \(k-\omega\) model for the 3D CFD simulations of the novel CMJICS. Moreover, the realizable \(k-\varepsilon\) turbulence model has been extensively validated for free, channel, boundary layer, and separated flows and resolves the turbulent round-jet/plane-jet spreading rate anomaly. The turbulent kinetic energy \((k)\) and eddy viscosity \((\mu_{t})\) are calculated by coupling the following transport equations for \(k\) and its dissipation rate \((\varepsilon)\), respectively,  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho k \overline{u}_{i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{ t}}{\sigma_{k}}\right)\frac{\partial k}{\partial x_{j}}\right]+G_{k}-\rho\varepsilon, \tag{9}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}}( \rho\varepsilon\overline{u}_{i})=\frac{\partial}{\partial x_{j}}\left[\left( \mu+\frac{\mu_{t}}{\sigma_{k}}\right)\frac{\partial\varepsilon}{\partial x_{j }}\right]+\rho C_{1}S\varepsilon-\rho C_{2}\frac{\varepsilon^{2}}{k+\sqrt{ \theta}\varepsilon}, \tag{10}\]The turbulent kinetic energy production rate appearing in Equation (9) is defined as \(G_{k}=\mu_{t}S^{2}\), where the modulus of the mean strain rate tensor is \(S=\sqrt{2\overline{S}_{ij}\overline{S}_{ij}}\). \(\mu_{t}\) is calculated as  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon}, \tag{11}\]  
where  
\[C_{\mu}=\frac{1}{A_{0}+A_{s}\frac{k\Omega t^{*}}{\varepsilon}}, \tag{12}\]  
where \(A_{0}=4.04\), \(A_{s}=\sqrt{6}\text{cos}\ \varphi\), where,  
\[\varphi=\frac{1}{3}\text{cos}^{-1}\left(\sqrt{6}W\right)\text{, }W=\frac{\overline{S}_{ij}\overline{S}_{jk}\overline{S}_{ki}}{\overset{ \sim}{S}},\ \overset{\sim}{S}=\sqrt{\overline{S}_{ij}\overline{S}_{ij}}, \tag{13}\]  
and  
\[U^{*}=\sqrt{\overline{S}_{ij}\overline{S}_{ij}+\overset{\sim}{\Omega_{ij} \Omega_{ij}}},\ \overset{\sim}{\Omega_{ij}}=\Omega_{ij}-2\varepsilon_{ijk}\omega_{k},\ \Omega_{ij}=\overline{\Omega}_{ij}- \varepsilon_{ijk}\Omega_{k}, \tag{14}\]  
where \(\overline{\Omega}_{ij}\) is the Reynolds-averaged rotation rate tensor viewed in a rotating reference frame with the angular velocity \(\omega_{k}\), and \(C_{1}=\text{max}\left(0.43,\ \frac{\eta}{\eta+5}\right)\) where, \(\eta=S\frac{k}{\varepsilon}\), and the model constants are specified as \(C_{2}=1.9,\ \sigma_{k}=1.0\) and \(\sigma_{\varepsilon}=1.2\). Both standard and realizable \(k-\varepsilon\) turbulence models are applicable to high \(Re\) number turbulent flows. In this study, the enhanced wall treatment for the \(\varepsilon\) equation available in ANSYS Fluent(r) (version 2019R1) is applied, which combines a two-layer model with enhanced wall functions to accurately solve for the viscosity-dominated near-wall region. As suggested by [70], the two-layer model smoothly blends the \(\mu_{t}\) determined by [PERSON]'s algebraic model in the near-wall region affected by viscosity [71] with its high Reynolds number definition in the outer region given by Equation (11). Similarly, \(\varepsilon\), which is determined algebraically in the viscosity-affected near-wall region and calculated by solving the transport equations in the outer region, is blended smoothly. The length scales for the \(\mu_{t}\) and \(\varepsilon\) appearing in the algebraic expressions are calculated according to the wall distance and a wall-distance-based turbulent \(Re\) number [72]. The blending function is determined as a transcendental function of the turbulent \(Re\) number [70].