## 2 Numerical Investigation  
The numerical analyses were performed on the inline and staggered tube banks, which are made up of 20 rows with tube diameters of \(D\) = 20 mm, and a longitudinal and transverse pitch \(S_{L}=S_{T}\) = 40 mm. The wall and inlet temperatures were constantly set to \(T_{w}\) = 333.15 K and \(T_{\infty}\) = 293.15 K, respectively, while the inlet velocities \(U_{\infty}\) corresponded to the Reynolds number in the range of \(10^{2}\)-\(10^{4}\). Due to a symmetrical character of the task, the domains shown in Figures 1 and 2 are considered. The flow is considered to be two-dimensional, steady, and incompressible. The governing equations of mass, momentum, and energy conservation are expressed in terms of the velocity vector \(v\), density \(\rho\), static pressure \(p\), gravity \(g\), dynamic viscosity \(\mu\), specific heat capacity \(c_{p}\), temperature \(T\), and thermal conductivity of fluid \(k_{f}\) as follows:  
\[\
abla\cdot\overrightarrow{v}=0 \tag{1}\]  
\[\rho\overrightarrow{v}\cdot\
abla\overrightarrow{v}=-\
abla p+\rho g+\mu \Delta\overrightarrow{v} \tag{2}\]  
\[\rho c_{p}\overrightarrow{v}\cdot\
abla T=k_{f}\Delta T \tag{3}\]  
The boundary conditions in terms of the velocities \(U\), \(V\) (x and \(y\) direction) are applied:  
\[\text{inlet: }U=U_{\infty},\ \ \ V=0,\ \ \ T=T_{\infty} \tag{4}\]  
\[\text{outlet}:\ P=P_{atm},\ \ \ T=T_{\infty} \tag{5}\]  
\[\text{tubes}:\ U=0,\ \ \ V=0,\ \ \ T=T_{w} \tag{6}\]  
\[\text{symmetry}:\ \ \frac{\partial U}{\partial y}=0,\ \ \ V=0,\ \ \ \frac{\partial T}{ \partial y}=0 \tag{7}\]  
Governing Equations (1)-(3) considering the boundary conditions (4)-(7) were solved by the finite volume method using the Ansys Fluent software. The steady pressure-based analyses with the variable air properties and incompressible ideal gas density model were performed. Regarding the variable properties, specific heat \(c_{p}\) (Jkg\({}^{-1}\)K\({}^{-1}\)), thermal conductivity \(k_{f}\) (Wm\({}^{-1}\)K\({}^{-1}\)), and dynamic viscosity \(\mu\) (kgm\({}^{-1}\)s\({}^{-1}\)) are considered as polynomial functions of temperature according to the standard [45]:  
\[c_{p}(T)=3.34\cdot 10^{-4}T^{2}-0.156T+1023.53 \tag{8}\]  
\[k_{f}(T)=-2.48\cdot 10^{-8}T^{2}+8.92\cdot 10^{-5}T+1.12\cdot 10^{-3} \tag{9}\]  
\[\mu(T)=-3.76\cdot 10^{-11}T^{2}+6.95\cdot 10^{-8}T+1.12\cdot 10^{-6} \tag{10}\]  
To perform the numerical analyses, the Direct Numerical Simulation solver was used for the laminar flow, while the turbulent one was solved by the k-\(\omega\) SST model, using a low inlet turbulent intensity. The pressure-velocity coupling was handled by the coupled scheme with the pseudo transient formulation. As a spatial discretization, the QUICK schemes were used. The solution was considered to be fully converged when the residuals of continuity, x-velocity, y-velocity, energy, k, and \(\omega\) parameters met the convergence criterion \(10^{-6}\). A preliminary study of the mesh grid size was carried out where quadrilateral elements were used. As it is shown in Table 1, five different numbers  
\[\rho\overrightarrow{v}\cdot\
abla\overrightarrow{v}=-\
abla p+\rho g+\mu \Delta\overrightarrow{v} \tag{2}\]0 The inline tube bank computational domain. \[\rho\overrightarrow{v}\cdot\
abla\overrightarrow{v}=-\
abla p+\rho g+\mu \Delta\overrightarrow{v} \tag{2}\]1 The staggered tube bank computational domain. of elements were created in the wall-adjacent area, and the average Nusselt number for the tube was evaluated. Each mesh variant has a different number and bias of elements in a normal direction to the tube wall in a distance of 1.3 mm, where a boundary layer is predicted. On the basis of this study, the last mesh variant with 14 perpendicular elements was chosen from the accuracy point of view. Furthermore, the \(y^{+}\) values are included in this study, with the worst values are shown in \[\rho\overrightarrow{v}\cdot\
abla\overrightarrow{v}=-\
abla p+\rho g+\mu \Delta\overrightarrow{v} \tag{2}\]2 The maximum element height out of the adjacent area is 0.3 mm.