# Modelling Pollutant Dispersion in Urban Canyons to Enhance Air Quality and Urban Planning
## 2 Materials and Methods
### CFD Model  
When conducting studies using Computational Fluid Dynamics (CFD) simulations, it is crucial to select some parameters that directly impact the accuracy of the results. Multiple configuration sets have been suggested to effectively address the turbulent nature of the problem [21; 36]. In this research, the chosen CFD software is Fluent 2023, a widely used tool in the scientific community due to its computational capabilities and versatility. To address the turbulent nature of the problem, Reynolds-averaged Navier-Stokes (RANS) equations have been employed, specifically using the k-\(\epsilon\) RNG closure scheme model, as suggested in different studies [34; 38] due to its good results in modelling the UBL. The incompressible Navier-Stokes equations in isothermal conditions are employed which consist of the continuity Equation (1),  
\[\frac{\partial\overline{u}_{i}}{\partial x_{i}}=0 \tag{1}\]  
And the momentum conservation Equation (2). \[\overline{u}_{i}\frac{\partial\overline{u}_{i}}{\partial x_{j}}=-\frac{ \partial\overline{p}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{(} v\frac{\partial\overline{u}_{i}}{\partial x_{j}}\Bigg{)}+\frac{\partial R_{ij}}{ \partial x_{j}} \tag{2}\]  
Both equations are written in tensor notation in which the indices \(i\) and \(j\) have a value of \(1\) and \(2\), respectively, due to the bidimensional nature of the problem. Because of this, \(x_{1}\) and \(x_{2}\) stand for the x-y cartesian coordinates, \(\overline{u}_{i}\) is the average air velocities in both  
Figure 1: Overview of the method. directions of space, \(\overline{p}\) is the average kinematic pressure, and \(\
u\) is the kinematic viscosity of the fluid. In addition, \(R_{ij}\) is the Reynolds stress tensor which is defined in Equation (3). \[R_{ij}=\
u_{t}\left(\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{ \partial\overline{u}_{j}}{\partial x_{i}}\right)-\frac{2}{3}\delta_{ij}k \tag{3}\]  
where \(\
u_{t}\) is the turbulent viscosity of the fluid, \(\delta_{ij}\) is delta Kronecker function, and \(k\) is the turbulent kinematic energy (TKE). In addition to these equations, \(k\)-\(\varepsilon\) renormalisation group model RNG adds two more equations, including one for TKE transport in Equation (4). \[\frac{\partial(k\overline{u}_{i})}{\partial x_{i}}=\frac{\partial}{\partial x _{i}}\left(\alpha_{k}\
u_{\text{eff}}\frac{\partial k}{\partial x_{i}}\right) +P_{k}-\varepsilon \tag{4}\]  
And the other is for TKE dissipation rate as in Equation (5). \[\frac{\partial(\varepsilon\overline{u}_{i})}{\partial x_{i}}=\frac{\partial} {\partial x_{i}}\left(\alpha_{\varepsilon}\
u_{\text{eff}}\frac{\partial \varepsilon}{\partial x_{i}}\right)+C_{\varepsilon 1}R_{k}\frac{\varepsilon}{ \overline{k}}-\left[C_{\varepsilon 2}+\frac{C_{\
u}\mu^{3}\left(1-\frac{\eta}{\eta_{0}} \right)}{1+\beta\eta^{3}}\right]\frac{\varepsilon^{2}}{k} \tag{5}\]  
where \(P_{k}\) is the kinematic turbulent energy production, \(\alpha_{k}\) and \(\alpha_{\varepsilon}\) are the inverse Prandtl numbers, and \(\
u_{\text{eff}}\) is the effective kinematic viscosity expressed as \(\
u_{\text{eff}}=\
u+\
u_{t}\) with turbulent viscosity obtained from \(\
u_{t}=C_{\
u}\frac{\xi^{2}}{\varepsilon}\) and \(\eta=Sk/\varepsilon\) where \(S\) is the scalar measure of the deformation tensor. Eventually, the turbulent production expression is in Equation (6). \[P_{k}=\
u_{t}\left(\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{ \partial\overline{u}_{j}}{\partial x_{i}}\right)\frac{\partial\overline{u}_{ i}}{\partial x_{j}} \tag{6}\]  
Finally, the species transport equation is shown in Equation (7).