## 2 Mathematical and Physical Formulation  
### Studying Problem and Governing Equations  
Figure 1 presents a schematic layout of a server room cooling system based on the cold aisle/hot aisle configuration. Multiple vertical rack units act as heat sources, each dissipating between 2 kW and 6 kW. These racks are arranged alternately, with 1.00 m and 1.20 m spacing, forming distinct cold aisles at the front and hot aisles at the rear. Cold air, produced by the CRAC unit, is distributed through a below-of-air duct located beneath the raised floor. It is then directed upwards via air vents, as shown by the blue arrows representing the cold air supply. This air enters the front of the rack units, absorbs heat generated by internal electronic components, and exits as hot air, indicated by the red arrows, through the rear of each rack. The use of directional arrows and clearly labeled elements (rack unit, air vent, air duct) enables a clear understanding of the thermal management strategy, highlighting the airflow paths and identifying the primary heat-generating elements within the system. This problem is considered to represent geometric and physical periodicity. Therefore, A single period will be considered for the analysis. The configuration under study is a rectangular cavity of size H, with its side walls exposed to periodic dynamic and thermal boundary conditions (Figure 2). To replicate heat sinks, two heat sources with a hot temperature \(T_{h}\) (\(T_{h}=50\)\({}^{\circ}\)C, operational temperature of the heat sink) are mounted on the cavity's bottom wall. Through two openings of length, convective fresh air is injected from the bottom of the cavity at a temperature of \(T_{c}\) (\(T_{c}=18\)\({}^{\circ}\)C, fresh air temperature). A long aperture is used on the upper horizontal wall to evacuate the hot air. Air, which is thought to be Newtonian and incompressible and for which the Prandtl number is \(P_{r}=0.71\), fills the enveloping. Periodic boundary conditions on the vertical walls are applied in order to represent a domain with an extended periodic structure and consistent geometric and flow patterns. This modeling choice is both computationally efficient and physically relevant, as it captures the essential transport mechanisms in systems where lateral effects are negligible. Byfocusing on a representative periodic unit, the simulation accurately reflects the core behavior of large-scale or modular systems such as finned heat exchangers, microchannel arrays, and porous media where periodicity governs overall performance. This approach ensures that the results are physically meaningful, reliable, and broadly applicable to practical thermal system designs. The following are the flow fields' macroscopic governing equations:  
\[\
abla.\mathbf{u}=0 \tag{1}\]  
\[\frac{\partial u}{\partial t}+(\mathbf{u}.\
abla)\mathbf{u}=-\frac{1}{\rho_{0}}.\
abla p +v\
abla^{2}\mathbf{u}+\mathbf{F} \tag{2}\]  
Figure 1: Detailed thermal configuration of the studied setup. where **F**, \(v\), **u**, p, \(\rho_{0}\) are the force field, kinematic viscosity, velocity, pressure, and reference density, in that order. The energy conservation equation is expressed as  
\[\frac{\partial T}{\partial t}+\textbf{{u}}.\
abla.T=\
abla.(\alpha.\
abla T)+S_{T} \tag{3}\]  
The scalar variable and diffusion coefficient for the incompressible thermal flows considered in this work are given as temperature \(T\) and thermal diffusivity \(\alpha\), respectively. The Bossiness assumption is used to include the impact of the temperature field on the flow field. The force field is defined as follows:  
\[\textbf{{F}}=-g\beta(T-T_{0})\textbf{{j}}+\textbf{{F}}_{\textbf{{v}}} \tag{4}\]  
where \(T_{0}\) is the reference temperature, \(j\) is the unit vector in the vertical direction, \(g\) is the gravitational acceleration magnitude, \(\textbf{{F}}_{v}\) is an external body force, and \(\beta\) is the thermal expansion coefficient.