# Aerodynamic Modification of High-Rise Buildings by the Adjoint Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equation

_{max}is the maximum line length of a cell, and ${F}_{1}$, ${F}_{2}$ are also the mixing functions of the shear stress transfer model [39]. The ${f}_{d}$ function that has been used is also a mixing function, and is in fact responsible for changing the models; it is defined as follows:

#### Adjoint Method

## 3. Computational Mesh and Numerical Verification

## 4. Result and Discussion

#### 4.1. Effects of Turbulence Models

#### 4.2. Investigation of Effective Parameters on the Aerodynamics of High-Rise Buiildings

## 5. Conclusions

- Investigating additional components of high-rise structures and examining their aerodynamic performance, considering elements like balconies, roofs, pedestrian areas, and others.
- Conducting a sensitivity analysis of the aforementioned geometric parameters (balconies length, width, …, and pedestrian displacement and etc.) in high-rise buildings, enabling the impact of each parameter to be ascertained.
- Employing heuristic and deterministic optimization techniques, including response surface methodology (RSM), genetic algorithms, and the adjoint method, to determine optimal solutions for the identified parameters.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${C}_{D}$ | Drag coefficient |

${C}_{L}$ | Lift coefficient |

CP | Mean pressure coefficient |

CFD | Computational fluid dynamic |

DDES | Delayed detached eddy simulation |

DES | Detached eddy simulation |

${d}_{w}$ | Distance from the wall surface |

K | Turbulent kinetic energy |

LES | Large eddy simulation |

${L}_{\mathrm{DDES}}$ | Length scale of the delayed detached eddy simulation |

${L}_{\mathrm{LES}}$ | Length scale of the large eddy simulation |

${L}_{\mathrm{RANS}}$ | Length scale of the Reynold average navier-stokes |

RANS | Reynold average navier-stokes |

Greek Symbols | |

φ | Wind flow direction |

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**Figure 5.**Comparison of pressure coefficient distribution at the front and behind various tall buildings in zero-degree wind flow direction, Re = 5.6 × 10

^{4}. (

**a**) the place of front and behind, (

**b**) the graph.

**Figure 6.**Comparison of pressure coefficient distribution at the front and behind two different triangular high-rise buildings at zero degree wind flow direction, Re = 5.6 × 10

^{4}. (

**a**) Front. (

**b**) Behind.

**Figure 7.**Turbulence intensity around buildings (

**a**) the triangular (

**b**) square buildings at zero degree wind flow direction, BL1.

**Figure 8.**Q-criterion distribution between two different buildings (

**a**) the triangular (

**b**) square buildings colored by streamwise x velocity at zero degree wind flow direction, BL1.

**Figure 9.**The iso surface x velocity of two different high-rise building configurations (

**a**) rectangle (

**b**) square at zero degree wind flow direction in BL1 condition.

**Figure 10.**Comparison of the drag coefficient of several buildings in different wind flow directions at BL1.

**Figure 11.**Comparison of sensitivity analysis of the aerodynamic coefficients of effective parameters.

**Figure 13.**Aerodynamic modification of a designed square section based on adjoint method. (

**a**) chamfer. (

**b**) roundness. (

**c**) single recession.

**Figure 14.**Power spectral density of lift coefficient; the aerodynamic modified buildings at zero-degree wind flow direction and BL1, (

**a**) chamfer, (

**b**) roundness, (

**c**) single recession.

**Figure 15.**Comparison of the streamlines around the buildings (

**a**) square, (

**b**) roundness oblong at zero-degree wind flow direction, BL1.

**Figure 16.**Sensitivity vector around the aerodynamic modified buildings at zero-degree wind flow direction and BL1, (

**a**) chamfer, (

**b**) roundness, (

**c**) single recession.

No Cases | Reynolds Number | Exponential Power-Low Factor | Wind Velocity | Turbulence Intensity |
---|---|---|---|---|

BL1-Flow condition | $5.6\times {10}^{4}$ | 0.24 | 8 | 19 |

BL2-Flow condition | $4.5\times {10}^{4}$ | 0.13 | 6.5 | 15 |

BL3-Flow condition | $3.5\times {10}^{4}$ | 0.07 | 5 | 12 |

**Table 2.**Comparison of drag coefficient according to different mesh cells around the square building in BL1.

Number of Cell | Drag Coefficient | Error (%) |
---|---|---|

548,000 | 0.986 | 5.2 |

2,880,000 | 1.024 | 1.5 |

5,270,000 | 1.11 | 6.9 |

**Table 3.**Comparison of drag coefficient between experimental results and numerical simulations for square high-rise building in various wind flow directions and (a) Re = 5.6 × 10

^{4}, (b) Re = 4.5 × 10

^{4}.

AOA | (a) BL1 | (b) BL2 | ||||
---|---|---|---|---|---|---|

Error (%) | ${\mathit{C}}_{\mathit{D}}$ (EXP) [42] | ${\mathit{C}}_{\mathit{D}}$ | Error (%) | ${\mathit{C}}_{\mathit{D}}$ (EXP) [42] | ${\mathit{C}}_{\mathit{D}}$ | |

0 | 1.5 | 1.04 | 1.024 | 1.4 | 1.19 | 1.173 |

15 | 4.8 | 0.9 | 0.943 | 7.6 | 0.98 | 1.054 |

30 | 11.7 | 0.92 | 0.812 | 8.3 | 0.99 | 0.908 |

45 | 11.1 | 0.94 | 1.044 | 9.8 | 1.02 | 1.120 |

Turbulence Model | ${\mathit{C}}_{\mathit{D}}$ (Experimental) = 1.04 [11] | |
---|---|---|

${\mathit{C}}_{\mathit{D}}$ (Numeric) | Error (%) | |

S A | 1.401 | 34.7 |

$k-\epsilon $ Standard | 1.245 | 22.6 |

$k-\epsilon $ RNG | 1.553 | 49.3 |

$k-\epsilon $ Realize | 1.425 | 37 |

$k-\omega $ Standard | 1.635 | 57.2 |

$k-\omega $ SST | 0.998 | 4 |

Reynolds Stress | 0.866 | 16.8 |

DES | 1.057 | 1.7 |

DDES | 1.024 | 1.5 |

Shape | Re = 5.6 × 10^{4}, BL1 | Re = 4.5 × 10^{4}, BL2 | Re = 3.5 × 10^{4}, BL3 |
---|---|---|---|

Square | 1.024 | 1.173 | 1.199 |

Circle | 0.423 | 0.4021 | 0.385 |

Rectangular | 1.176 | 1.230 | 1.268 |

Triangular | 0.871 | 0.846 | 0.84 |

**Table 6.**Lift and drag coefficients loaded on the square and oblong buildings at BL1, zero-degree wind flow direction.

Shape | Lift Coefficient | Drag Coefficient |
---|---|---|

Square | −0.0004 | 1.024 |

Chamfer | −0.0033 | 0.783 |

Roundness | 0.0068 | 0.7044 |

Single recession | −0.0065 | 0.745 |

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**MDPI and ACS Style**

Nikkhoo, A.; Esmaeili, A.; Rabizade, S.; Zamiri, M.
Aerodynamic Modification of High-Rise Buildings by the Adjoint Method. *J* **2024**, *7*, 72-93.
https://doi.org/10.3390/j7010004

**AMA Style**

Nikkhoo A, Esmaeili A, Rabizade S, Zamiri M.
Aerodynamic Modification of High-Rise Buildings by the Adjoint Method. *J*. 2024; 7(1):72-93.
https://doi.org/10.3390/j7010004

**Chicago/Turabian Style**

Nikkhoo, Amirfarhang, Ali Esmaeili, Shayan Rabizade, and Majid Zamiri.
2024. "Aerodynamic Modification of High-Rise Buildings by the Adjoint Method" *J* 7, no. 1: 72-93.
https://doi.org/10.3390/j7010004