# Wind Loading on Scaled Down Fractal Tree Models of Major Urban Tree Species in Singapore

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Tree Models

#### 2.1.1. Development of the Fractal Tree Models

#### 2.1.2. Development of the Tree Crown Models

#### 2.2. Wind Tunnel Tests

#### 2.2.1. Wake Profile Measurement

#### 2.2.2. Bulk Drag Measurement

- ${A}_{ref}$ = Reference area;
- ${C}_{D}$ = Bulk drag coefficient;
- $D$ = Bulk drag measurement in wind tunnel;
- ${U}_{ref}$ = Reference wind speed;
- $\rho $ = Air density.

#### 2.3. Numerical Simulations

#### 2.3.1. Solver and Numerical Models

- $\mathrm{FSAD}$ = Frontal silhouette area density, frontal silhouette area divided by tree crown volume;
- ${S}_{{u}_{i}}$ = Momentum sink;
- ${C}_{d}$ = Drag coefficient;
- $U$ = Velocity magnitude = $\sqrt{{u}_{i}{u}_{i}}$ (using the Einstein summation convention);
- ${u}_{i}$ = Velocity component.

#### 2.3.2. Grid and Boundary Conditions

^{−4}s. The residual convergence criteria used in this study was 10

^{−6}. It measures the iterative solution’s convergence and directly quantifies the error in the solution of the system of equations. The simulations were initialized until the velocity and pressure field data were statistically stable. In the current work, statistical stability is reached when the difference in a time-averaged quantity is less than 0.1% when the averaging period is halved. In this study, the mean drag and velocity profiles at 1H and 2H are examined to ensure the solutions are statistically stable.

#### 2.3.3. Tree Modelling

- ${A}_{total}$ = Total frontal optical silhouette area;
- ${A}_{fs}$ = Frontal optical silhouette area;
- ${C}_{d}$ = Local drag coefficient;
- $D$ = Drag force measurement in wind tunnel;
- $N$ = Total number of discretized elements;
- ${V}_{e}$ = Volume of element.

^{3}, 10

^{3}, and 20

^{3}elements, namely 5, 10, and 20 elements in the directions of height, width, and depth respectively. This resulted in 125, 1000, and 8000 total elements for each respective resolution. The 9th and 20th slices for K. senegalensis tree model are presented in Figure 7. In the 9th slice, the branches show up near to the trunk of the tree model. In the final slices, the tree crown branches no longer exist at the outermost point of the tree crown. The frontal silhouette area of each slice is presented in black. The silhouette area is the product of the number of black pixels and the area represented by each pixel. The summation of frontal silhouette area in all slices is the total frontal silhouette area; which was used as the reference area in the calculation of the local drag coefficient. The resultant local ${C}_{d}\xb7\mathrm{FSAD}$ is presented in Figure 7d. The trunk has the greatest local value. The streamwise averaged and longitudinal averaged coefficient of K. senegalensis at different resolutions are plotted in Figure 8. In the tree model volume, the center of the crown has the largest values. Similar plots for other tree models are presented in Figure A7 in Appendix B.

## 3. Results and Discussion

#### 3.1. Velocity Profile

^{3}increased by 2.5 times compared to Δ = 10

^{3}. As seen in the figure, the results at Δ = 10

^{3}and Δ = 20

^{3}are very close. We hence chose to simulate most of the cases at a resolution of Δ = 10

^{3}, except for P. pterocarpum and S. saman, which were simulated at Δ = 20

^{3}due to the significantly wider crowns.

#### 3.2. Drag Force

#### 3.3. Bulk Drag Coefficients

- $\alpha $ = Aerodynamic porosity;
- $\beta $ = Frontal optical porosity.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**H. odorata fractal tree model: (

**a**) trunk and branches; (

**b**) tree crown volume; (

**c**) final model (frontal view at 0° rotation).

**Figure A2.**S. saman fractal tree model: (

**a**) trunk and branches; (

**b**) tree crown volume; (

**c**) final model (frontal view at 0° rotation).

**Figure A3.**S. grande fractal tree model: (

**a**) trunk and branches; (

**b**) tree crown volume; (

**c**) final model (frontal view at 0° rotation).

**Figure A4.**S. macrophylla fractal tree model: (

**a**) trunk and branches; (

**b**) tree crown volume; (

**c**) final model (frontal view at 0° rotation).

**Figure A5.**T. rosea fractal tree model: (

**a**) trunk and branches; (

**b**) tree crown volume; (

**c**) final model (frontal view at 0° rotation).

**Figure A6.**P. pterocarpum fractal tree model: (

**a**) trunk and branches; (

**b**) tree crown volume; (

**c**) final model (frontal view at 0° rotation).

## Appendix B

**Figure A7.**Streamwise averaged ${C}_{d}\xb7\mathrm{FSAD}$ for 0° rotation angle: (

**a**) H. odorata (Δ = 10

^{3}); (

**b**) S. saman (Δ = 20

^{3}); (

**c**) S. grande (Δ = 10

^{3}); (

**d**) S. macrophylla (Δ = 10

^{3}); (

**e**) T. rosea (Δ = 10

^{3}); (

**f**) P. pterocarpum (Δ = 20

^{3}).

## Appendix C

**Figure A8.**Wake contour at 1H downstream for 0° rotation angle: (

**a**) H. odorata; (

**b**) S. saman; (

**c**) S. grande; (

**d**) S. macrophylla; (

**e**) T. rosea; (

**f**) P. pterocarpum.

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**Figure 1.**Fabricated tree models: (

**a**) Khaya senegalensis; (

**b**) Hopea odorata; (

**c**) Samanea saman; (

**d**) Swietenia macrophylla; (

**e**) Syzygium grande; (

**f**) Tabebuia rosea; (

**g**) Peltophorum pterocarpum. K. senegalensis, and H. odorata were painted black to reduce laser reflections in particle image velocimetry (PIV) measurements.

**Figure 2.**K. senegalensis fractal tree model: (

**a**) trunk and branches; (

**b**) tree crown volume; (

**c**) final model (frontal view at 0° rotation).

**Figure 3.**K. senegalensis: (

**a**) photo of real tree; (

**b**) binary image; (

**c**) image with defined crown area.

**Figure 7.**(

**a**) K. senegalensis tree model resolution Δ = 20

^{3}; (

**b**) the 9th slice; (

**c**) the 20th slice; (

**d**) resultant local ${C}_{d}\xb7\mathrm{FSAD}$.

**Figure 8.**Averaged ${C}_{d}\xb7\mathrm{FSAD}$ of K. senegalensis at 0° rotation $\left(\frac{\int \text{}{C}_{d}\xb7\mathrm{FSAD}dy}{{\mathsf{\Delta}}^{\frac{1}{3}}}\right)$: averaged along the streamwise direction (

**a**) Δ = 5

^{3}; (

**b**) Δ = 10

^{3}; (

**c**) Δ = 20

^{3}; (

**d**) further averaged along the x-direction $\left(\frac{\iint {C}_{d}\xb7\mathrm{FSAD}dydx}{{\mathsf{\Delta}}^{\frac{2}{3}}}\right)$.

**Figure 9.**Normalized average horizontal velocity contour (${u}_{y}/{U}_{\mathrm{ref}}$) of K. senegalensis at ${U}_{\mathrm{ref}}$ = 5 m/s and 0° rotation as measured from the wind tunnel experiment.

**Figure 10.**Simulated wake contours $({u}_{y}/{U}_{\mathrm{ref}})$ behind the tree model at (

**a**) Y = 0H; (

**b**) Y = 1H; (

**c**) Y = 2H.

**Figure 11.**Average horizontal velocity profile at 1H upstream (

**left**), 1H downstream (

**middle**), and 2H downstream (

**right**) of K. senegalensis 0° rotation, 15 m/s upstream wind speed.

**Figure 13.**Average horizontal velocity profile at 1H upstream (

**left**), 1H downstream (

**middle**), and 2H downstream (

**right**): (

**a**) K. senegalensis 0° rotation; (

**b**) K. senegalensis 45° rotation; (

**c**) K. senegalensis 90° rotation; (

**d**) H. odorata 0° rotation; (

**e**) H. odorata 45° rotation; (

**f**) H. odorata 90° rotation.

**Figure 14.**Bulk drag coefficient against frontal optical porosity of the tree models. Data of the specified flexible models/trees are data extracted from Manickathan et al. [22]. The reference area is the frontal crown area.

Tree Species | Sample Size | Optical Porosity | |
---|---|---|---|

Average | Standard Deviation | ||

K. senegalensis | 5 | 0.1237 | 0.0711 |

H. odorata | 5 | 0.1005 | 0.0732 |

S. saman | 7 | 0.1864 | 0.0743 |

S. macrophylla | 11 | 0.1424 | 0.0506 |

S. grande | 7 | 0.1157 | 0.0573 |

T. rosea | 12 | 0.1172 | 0.0421 |

P. pterocarpum | 7 | 0.1583 | 0.0499 |

K. senegalensis at 0° Rotation | ||||
---|---|---|---|---|

Experiment | Simulations | |||

Δ = 5^{3} | Δ = 10^{3} | Δ = 20^{3} | ||

Drag, N | 2.040 ± 0.016 | 1.90 | 2.00 | 2.02 |

Difference, % | −6.75 | −2.03 | −0.86 |

Tree Species | Rotation Angle, ° | Drag, N | Difference, % | |
---|---|---|---|---|

Experiment | Simulations | |||

K. senegalensis | 0 | 2.040 ± 0.016 | 2.00 | −2.03 |

45 | 2.022 ± 0.007 | 2.11 | 4.18 | |

90 | 1.896 ± 0.007 | 2.00 | 5.41 | |

H. odorata | 0 | 2.402 ± 0.011 | 2.35 | −2.00 |

45 | 2.286 ± 0.016 | 2.28 | −0.08 | |

90 | 2.262 ± 0.013 | 2.27 | 0.23 | |

S. saman^{1} | 0 | 3.116 ± 0.039 | 3.13 | 0.34 |

45 | 3.080 ± 0.041 | 3.27 | 6.01 | |

90 | 2.745 ± 0.012 | 2.88 | 4.96 | |

S. macrophylla | 0 | 1.842 ± 0.025 | 1.87 | 1.43 |

45 | 1.700 ± 0.011 | 1.79 | 5.17 | |

90 | 1.526 ± 0.009 | 1.54 | 1.12 | |

S. grande | 0 | 1.749 ± 0.016 | 1.72 | −1.56 |

45 | 1.608 ± 0.013 | 1.64 | 1.77 | |

90 | 1.658 ± 0.014 | 1.72 | 3.94 | |

T. rosea | 0 | 1.969 ± 0.012 | 1.92 | −2.49 |

45 | 1.584 ± 0.017 | 1.57 | −1.14 | |

90 | 1.056 ± 0.007 | 1.04 | −1.14 | |

P. pterocarpum^{1} | 0 | 4.288 ± 0.022 | 4.65 | 8.33 |

45 | 3.814 ± 0.044 | 4.09 | 7.20 | |

90 | 3.355 ± 0.016 | 3.74 | 3.93 |

^{1}Simulated at Δ = 20

^{3}.

Tree Species | Wind Speed, m/s | ${\mathit{C}}_{\mathit{D}}$ | Tree Species | Wind Speed, m/s | ${\mathit{C}}_{\mathit{D}}$ |
---|---|---|---|---|---|

Populus trichocarpa^{1} | 5 | 0.780 | Thuja plicata^{2} | 5 | 0.886 |

10 | 0.642 | 10 | 0.696 | ||

15 | 0.574 | 15 | 0.596 | ||

Populus tremuloides^{1} | 5 | 0.817 | Hibiscus syriacus^{3} | 5 | 0.607 |

10 | 0.688 | 10 | 0.531 | ||

15 | 0.647 | 15 | 0.491 | ||

Alnus rubra^{1} | 5 | 0.738 | Thuja occidentalis^{3} | 5 | 0.856 |

10 | 0.595 | 10 | 0.791 | ||

15 | 0.551 | 15 | 0.753 | ||

Betula papyrifera^{1} | 5 | 0.765 | Ilex crenata^{3} | 5 | 0.807 |

10 | 0.660 | 10 | 0.780 | ||

15 | 0.640 | 15 | 0.765 | ||

Acer macrophyllum^{1} | 5 | 0.813 | S. grande^{4} | 1 | 1.521 |

10 | 0.635 | 2 | 0.509 | ||

15 | 0.599 | 3 | 0.319 | ||

Tsuga heterophylla^{2} | 5 | 1.117 | K. senegalensis^{4} | 1 | 1.565 |

10 | 1.030 | 2 | 0.506 | ||

15 | 0.941 | 3 | 0.390 | ||

Pinus contorta^{2} | 5 | 1.037 | |||

10 | 0.940 | ||||

15 | 0.836 |

^{1}Hardwood saplings in wind tunnel. ${C}_{D}$ calculated using wind-speed-specific frontal area [18].

^{2}Conifer saplings in wind tunnel. ${C}_{D}$ calculated using wind-speed-specific frontal area [19].

^{3}Real trees in wind tunnel. ${C}_{D}$ calculated using wind-speed-specific frontal area [21].

^{4}Real mature trees in field tests. ${C}_{D}$ calculated using still air frontal area [37].

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**MDPI and ACS Style**

Chan, W.-L.; Eng, Y.; Ge, Z.; Lim, C.W.C.; Gobeawan, L.; Poh, H.J.; Wise, D.J.; Burcham, D.C.; Lee, D.; Cui, Y.;
et al. Wind Loading on Scaled Down Fractal Tree Models of Major Urban Tree Species in Singapore. *Forests* **2020**, *11*, 803.
https://doi.org/10.3390/f11080803

**AMA Style**

Chan W-L, Eng Y, Ge Z, Lim CWC, Gobeawan L, Poh HJ, Wise DJ, Burcham DC, Lee D, Cui Y,
et al. Wind Loading on Scaled Down Fractal Tree Models of Major Urban Tree Species in Singapore. *Forests*. 2020; 11(8):803.
https://doi.org/10.3390/f11080803

**Chicago/Turabian Style**

Chan, Woei-Leong, Yong Eng, Zhengwei Ge, Chi Wan Calvin Lim, Like Gobeawan, Hee Joo Poh, Daniel Joseph Wise, Daniel C. Burcham, Daryl Lee, Yongdong Cui,
and et al. 2020. "Wind Loading on Scaled Down Fractal Tree Models of Major Urban Tree Species in Singapore" *Forests* 11, no. 8: 803.
https://doi.org/10.3390/f11080803