A multi-model study of the air pollution related to traffic flow in a two-dimensional porous metropolitan area

TL;DR

This work addresses urban air pollution from traffic by treating the city as a two-dimensional porous medium and coupling a nonconservative macroscopic traffic-flow model with a Darcy-Brinkman-Forchheimer momentum equation to a microscopic CO$_2$ emission model and a convection-diffusion-reaction transport for dispersion. A unified finite-element framework using P1 elements discretizes the coupled systems on a common mesh, with explicit time stepping for the flow fields and a stabilized GLS scheme for the transport equation. Numerical experiments in the Guadalajara Metropolitan Area compare dense and disperse city porosities, revealing strong effects of porosity on traffic speed, wind patterns, and pollutant evacuation, as well as the mitigating impact of imposed speed limits on mean CO$_2$ concentrations. The results demonstrate a scalable, physically grounded tool for analyzing urban pollution and guiding mitigation strategies in two-dimensional porous-city settings.

Abstract

In this paper, a useful reinterpretation of the city as a porous medium justifies the application of well-known models on fluid dynamics to develop a multi-model study of urban air pollution due to traffic flow in a large city. Thus, to simulate the traffic flow through the city we use a nonconservative macroscopic traffic model combining the continuity equation with the Darcy-Brinkman-Forchheimer equations. For the air flow, regarding the emission rate of CO$_2$ and its dispersion in the atmosphere, we combine a microscopic model -- based on regression techniques but depending on vehicles' velocity and acceleration -- with a classical convection-diffusion-reaction transport model. To solve numerically above PDEs models, the finite element method of Lagrange $\rm{P_1}$ type along with suitable time marching schemes (like the strong stability preserving scheme) were sufficient to obtain stable numerical solutions. Several computational tests were run on a realistic scenario inspired by the Metropolitan Area of Guadalajara (Mexico), showing not only the influence of the urban landscape (that is, the porosity) on traffic flow, air flow, and pollution transport, but also other interesting phenomena such as rarefaction traffic waves.

Figures (11)

• Figure 1: (a) The Metropolitan Area of Guadalajara and its rural surroundings, taken from Google-Earth (2024), and (b) the triangular mesh generated with the software Gmsh, showing the numbers for the different zones and the labels of the boundary segments.
• Figure 2: Porosity distribution in the domain $\Omega$ for the two city configurations: (a) The dense buildings city, with a porosity value of 0.38 at the center and a value of 0.82 on the city limits. (b) The disperse city, showing a 0.68 porosity value at its center and similar values to the dense case on suburbia.
• Figure 3: (a) Initial condition for the vehicular density. (b) Absorption rate of the solid phase (off-street parking) for the dense city. (c) Distribution of the eikonal potential $\varphi$ on the dense city. (d) Desired traffic speed at initial time $t=0$.
• Figure 4: Vehicular density $\rho$$[\rm{veh/km^2}]$ for the dense city (left column) and the disperse city (right column) at times 0.5, 1.0, 1.5 h (from top to bottom).
• Figure 5: Traffic local velocity $\mathbf{u}$$[\rm{km/h}]$ for the dense city (left column) and the disperse city (right column) at times 0.5, 1.0, 1.5 h (from top to bottom).