# Urban Heat Island Dynamics in an Urban–Rural Domain with Variable Porosity: Numerical Methodology and Simulation

^{1}

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## Abstract

**:**

_{1}type combined with explicit and implicit time-marching schemes have been effective for high-quality numerical simulations. Several numerical tests were performed on a domain inspired by the metropolitan region of Guadalajara (Mexico), in which not only the temperature inversion was reproduced but also the simulation of the UHI transition by strong wind gusts.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Computer Fluid Dynamics and Heat Transfer Models on Porous Media

#### 2.2. A Non-Stationary Darcy–Brinkman–Forchheimer Model for an Urban–Rural Porous Media

#### 2.3. Model for Heat Transfer in Urban–Rural Domain

**Remark**

**1.**

**Remark**

**2.**

#### 2.4. Numerical Solution

**Remark**

**3.**

#### 2.4.1. An Explicit Scheme for the Darcy–Forchheimer–Brinkman Equation: The Chorin Method

**Remark**

**4.**

#### 2.4.2. A Finite Element Approach and Implicit Time Schemes to Solve the Heat-Transfer Model

^{™}software was used to implement the finite element method and, particularly, its command

`fsolve`was used to address the system of non-linear equations. This solver allowed us to supply the gradient (23a,b) and optionally evaluate its accuracy by comparing them with the solver’s gradients (estimated with central finite differences). This comparison proved to be effective, with a maximum error of order $\mathcal{O}\left({10}^{-5}\right)$ and $\mathcal{O}\left({10}^{-6}\right)$ for (23a) and (23b), respectively.

## 3. Numerical Results

#### 3.1. Parameters Values and the Urban–Rural Domain Based on the Metropolitan Region of Guadalajara

**Remark**

**5.**

#### 3.2. A 24 h Simulation of the UHI under Ideal Conditions

**Remark**

**6.**

#### 3.3. Influence of the Wind and the Need for Numerical Stabilization

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Symbol/Formula | Definition | Value/Units |
---|---|---|

${R}_{s}$ | Global solar radiation | W m${}^{-2}$ |

$\rho $ | Fluid density | kg m${}^{-3}$ |

${\rho}_{a}$ | Air density | 1.1614 kg m${}^{-3}$ |

${\rho}_{urb}$ | Urban soil density | 2.11 × 10${}^{3}$ kg m${}^{-3}$ |

${\rho}_{rural}$ | Rural soil density | 8.4 × 10${}^{2}$ kg m${}^{-3}$ |

${c}_{a}$ | Specific heat of air | 1005 J kg${}^{-1}$ K${}^{-1}$ |

${c}_{urb}$ | Specific heat of urban soil | 920 J kg${}^{-1}$ K${}^{-1}$ |

${c}_{rural}$ | Specific heat of rural soil | 3600 J kg${}^{-1}$ K${}^{-1}$ |

${c}_{steam}$ | Specific heat of steam | 1952 J kg${}^{-1}$ K${}^{-1}$ |

${\alpha}_{air}$ | Air conductivity | 0.0263 J s${}^{-1}$ m${}^{-1}$ K${}^{-1}$ |

${\alpha}_{urb}$ | Urban soil conductivity | 0.41 J s${}^{-1}$ m${}^{-1}$ K${}^{-1}$ |

${\alpha}_{rural}$ | Rural soil conductivity | 1.47 J s${}^{-1}$ m${}^{-1}$ K${}^{-1}$ |

${h}_{a}$ | Air convection coefficient | 1 J s${}^{-1}$ m${}^{-2}$ K${}^{-1}$ |

${h}_{urb}$ | Urban soil convection coefficient | 0.4 J s${}^{-1}$ m${}^{-2}$ K${}^{-1}$ |

${h}_{rural}$ | Rural soil convection coefficient | 0.2 J s${}^{-1}$ m${}^{-2}$ K${}^{-1}$ |

${a}_{urb}$ | Urban albedo | $0.27$ |

${a}_{rural}$ | Rural albedo | $0.16$ |

${e}_{urb}$ | Urban soil emissivity | $0.96$ |

${e}_{rural}$ | Rural soil emissivity | $0.85$ |

${e}_{sky}$ | Sky emissivity | $0.77$ |

${z}_{0,urb}$ | Urban soil roughness | 7 m |

${z}_{0,rural}$ | Rural soil roughness | 1 m |

${u}_{*,urb}$ | Urban friction velocity | 0.2 ms${}^{-1}$ |

${u}_{*,rural}$ | Rural friction velocity | 0.5 ms${}^{-1}$ |

${\beta}_{urb}$ | Urban Bowen radius | $5.0$ |

${\beta}_{rural}$ | Rural Bowen radius | $0.5$ |

${d}_{a}=2$ | Air layer thickness | 2.0 m |

${d}_{s}=1$ | Soil layer thickness | 1.0 m |

k | Von Karman constant | $0.4$ |

$Nu$ | Nusselt number | 1 |

${\sigma}_{B}$ | Stephan–Boltzmann constant | 5.6703 × 10${}^{-8}$ W m${}^{-2}$ K${}^{-4}$ |

r | Urban radius | 13,250.0 m |

$({x}_{c},{y}_{c})$ | Urban center coordinates | (0, −2500) m |

d | Diameter of spheres | 1 m |

${C}_{F}=\frac{175}{\sqrt{150\phantom{\rule{0.166667em}{0ex}}{\u03f5}^{3}}};$ | Forchheimer coefficient | $--$ |

$({\alpha}_{x},{\alpha}_{y})$ | Gaussian distribution variances | (10${}^{-8.25}$, 10${}^{-8.15}$) |

$K=\frac{{\u03f5}^{3}\phantom{\rule{0.166667em}{0ex}}{d}^{2}}{150{(1-\u03f5)}^{2}}$ | Permeability | $--$ |

${r}_{sh}=\frac{0.75\phantom{\rule{0.166667em}{0ex}}{\rho}_{a}\phantom{\rule{0.166667em}{0ex}}{c}_{steam}}{{\alpha}_{a}\phantom{\rule{0.166667em}{0ex}}Nu}$ | Soil resistance | s m${}^{-1}$ |

${r}_{ah}=\frac{1}{{k}^{2}\phantom{\rule{0.166667em}{0ex}}{u}^{*}}{\left[ln\left(\frac{2}{{z}_{0}}\right)\right]}^{2}$ | Air resistance | s m${}^{-1}$ |

${\sigma}_{a}=\frac{{\sigma}_{B}\phantom{\rule{0.166667em}{0ex}}{e}_{s}}{{\rho}_{s}\phantom{\rule{0.166667em}{0ex}}{c}_{s}}$ | Air radiation interchange coefficient | m s${}^{-1}$ K${}^{-3}$ |

${\gamma}_{a}=\frac{{h}_{a}}{{c}_{a}\phantom{\rule{0.166667em}{0ex}}{\rho}_{a}}$ | Air convective interchange coefficient | m s${}^{-1}$ |

${\gamma}_{s}=\frac{{h}_{s}}{{c}_{s}\phantom{\rule{0.166667em}{0ex}}{\rho}_{s}}$ | Soil convective interchange coefficient | m s${}^{-1}$ |

${\mu}_{a}=\frac{{\alpha}_{a}}{{c}_{a}\phantom{\rule{0.166667em}{0ex}}{\rho}_{a}}$ | Air thermal diffusivity | m${}^{2}$s${}^{-1}$ |

${\mu}_{s}=\frac{{\alpha}_{s}}{{\rho}_{s}\phantom{\rule{0.166667em}{0ex}}{c}_{s}}$ | Soil thermal diffusivity | m${}^{2}$s${}^{-1}$ |

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**Figure 1.**Heat-transfer model: the urban–rural domain $\Omega $ (bottom) is a sub-set of ${\mathbb{R}}^{2}$ with an urban area, represented as a rectangle (gray); a rural area (green); the soil below (brown); inlet–outlet boundary segments; and inlet wind direction (blue arrow). Based on the horizontal section of the domain (magenta sector), the vertical distribution of the three temperatures ${\theta}_{a}$, ${\theta}_{0}$, and ${\theta}_{s}$ is shown (top left) in congruence with the vertical interchange thermal resistances ${r}_{ah}$ and ${r}_{sh}$ for air and soil, respectively (top right). Both layer thicknesses, ${d}_{a}$ for air and ${d}_{s}$ for soil, are much smaller when compared with the horizontal scale. Finally, the resistance diagram (top right) is formulated based on Figure 6 in [23].

**Figure 2.**Study area idealization, mesh, and 24 h solar radiation. (

**a**) Satellite photograph of the metropolitan region of Guadalajara with more of 40 km${}^{2}$ (Google-Earth, 2023). Here, we denoted the boundaries and hills (idealized as rectangles) of the domain $\Omega $ with a red line. (

**b**) Triangular mesh ${\tau}_{h}$, inlet boundary ${\Gamma}_{in}$ (green), outlet boundary ${\Gamma}_{out}$ (magenta), hills walls ${\Gamma}_{w}$ (red), and the urban-limit layout (black) as a circumference of radius r. The black arrow shows the inlet wind direction. (

**c**) Typical global day-long solar radiation ${R}_{s}\left(t\right)$ with maximum values at midday and almost zero on either side.

**Figure 3.**Examples of parameter distribution on the urban–rural domain and the urban–limit layout. (

**a**) Gaussian distribution of the porosity $\u03f5$ with lower values at the city center and with higher values in the rural surroundings. (

**b**) For dimensionless parameters such as soil emissivity ${e}_{s}$, a simple radial distribution with only two values, corresponding to urban and rural, was assigned to each mesh node.

**Figure 4.**Time evolution of the air temperatures during a 24 h interval. We used the following times $8,\phantom{\rule{0.166667em}{0ex}}12,\phantom{\rule{0.166667em}{0ex}}16,\phantom{\rule{0.166667em}{0ex}}$ and 20 h as a sample to illustrate the obtained results. (

**a**) ${\theta}_{a}$ at 8 h. (

**b**) ${\theta}_{a}$ at 10 h. (

**c**) ${\theta}_{a}$ at 12 h. (

**d**) ${\theta}_{a}$ at 14 h. (

**e**) ${\theta}_{a}$ at 16 h. (

**f**) ${\theta}_{a}$ at 20 h.

**Figure 5.**The reference wind field used in the experiment was constant and computed from the momentum equation. (

**a**) Wind field. (

**b**) Norm of the wind field $\parallel \mathbf{u}\parallel $.

**Figure 6.**It was clear that the streamline diffusion stabilizer minimized the spurious oscillation without notable changes in the temperatures. (

**a**) Un-stabilized solution ($\zeta =0$). (

**b**) Stabilized solution ($\zeta =25$).

**Figure 7.**Wind effect on the UHI. As expected, the wind transported the warmer air from urban to rural areas. With enough intensity, the wind could be a factor in cold downwind air temperatures. (

**a**) Reference wind $\eta =1$. (

**b**) Augmented reference wind $\eta =2$. (

**c**) Augmented reference wind $\eta =3$. (

**d**) Augmented reference wind $\eta =4$.

**Figure 8.**Dynamics of the transport of warmer mass air during wind gusts. (

**a**) ${\theta}_{a}$ before wind blows at 13 h. (

**b**) ${\theta}_{a}$ an in-between time period of blowing wind. (

**c**) ${\theta}_{a}$ final state at 17 h.

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## Share and Cite

**MDPI and ACS Style**

García-Chan, N.; Licea-Salazar, J.A.; Gutierrez-Ibarra, L.G.
Urban Heat Island Dynamics in an Urban–Rural Domain with Variable Porosity: Numerical Methodology and Simulation. *Mathematics* **2023**, *11*, 1140.
https://doi.org/10.3390/math11051140

**AMA Style**

García-Chan N, Licea-Salazar JA, Gutierrez-Ibarra LG.
Urban Heat Island Dynamics in an Urban–Rural Domain with Variable Porosity: Numerical Methodology and Simulation. *Mathematics*. 2023; 11(5):1140.
https://doi.org/10.3390/math11051140

**Chicago/Turabian Style**

García-Chan, Néstor, Juan A. Licea-Salazar, and Luis G. Gutierrez-Ibarra.
2023. "Urban Heat Island Dynamics in an Urban–Rural Domain with Variable Porosity: Numerical Methodology and Simulation" *Mathematics* 11, no. 5: 1140.
https://doi.org/10.3390/math11051140