## 2 Methodology  
In this section, the mathematical formulations describing the air flow, heat transfer and the model of turbulence used for this study are presented. When analyzing any air flow problem, it is necessary to establish the conservation laws that are represented mathematically and physically by the governing equations. The governing equations are composed of the state equation, continuity equation, momentum equation, and energy equation. In this paper, a three-dimensional case is considered, where the working fluid is assumed to be an ideal gas. The air flow regime is assumed to be turbulent and the heat transfer process is assumed to be steady. The state equation is a constitutive relation of a fluid which relates several properties. In engineering, the substances of interest are mostly gases, with moderate pressures and temperatures. In this paper, as mentioned above, an ideal gas is used, and the equation of state is given as follows [37]:  
\[\rho=\frac{pM_{m}}{R_{u}T} \tag{1}\]  
in which \(\rho\) is the specific mass and is given in kg\(\cdot\)m\({}^{-3}\), the pressure is given by \(p\) and it is measured in Pa, \(M_{m}\) represents the molar mass of the gas and it is measured in kg\(\cdot\)kmol\({}^{-1}\), the ideal gas constant is represented by \(R_{u}\) and it is measured in J\(\cdot\)mol\({}^{-1}\cdot\)K\({}^{-1}\), and the temperature \(T\) is given in K. Modeling the air as an ideal gas allows taking into account the buoyancy effect. The continuity equation, also referred to as the mass conservation equation, expresses that the mass entering a control volume is equal to the mass exiting that control volume. Consequently, no losses occur in this process. The continuity equation is expressed as follows [3]:  
\[\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}+\frac {\partial(\rho w)}{\partial z}+\frac{\partial\rho}{\partial t}=0 \tag{2}\]  
where \(t\) is the time in seconds, \(x\), \(y\) and \(z\) are the Cartesian coordinates and \(u\), \(v\) and \(w\) are the corresponding velocity components in m\(\cdot\)s\({}^{-1}\). The equation of momentum can be obtained through [PERSON]'s second law, which asserts that the outcomes of the forces applied to a particle is equal to the rate of change of its linear momentum. These equations are presented as follows [37]:  
\[\rho\bigg{(}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\bigg{)}=-\frac{ \partial p}{\partial x}+\mu\bigg{(}\frac{\partial^{2}u}{\partial x^{2}}+\frac {\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\bigg{)} \tag{3}\]  
\[\rho\bigg{(}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\bigg{)}=-\frac{ \partial p}{\partial y}+\mu\bigg{(}\frac{\partial^{2}v}{\partial x^{2}}+\frac {\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}v}{\partial z^{2}}\bigg{)} \tag{4}\]  
\[\rho\bigg{(}\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v \frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\bigg{)}=\rho g- \frac{\partial p}{\partial z}+\mu\bigg{(}\frac{\partial^{2}w}{\partial x^{2} }+\frac{\partial^{2}w}{\partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}} \bigg{)} \tag{5}\]  
where \(\mu\) is the dynamic viscosity in kg\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\) and \(g\) is the gravitational acceleration in m\(\cdot\)s\({}^{-2}\). The gravitational force term is included in the \(z\) momentum equation.