An Innovative Method for Wind Load Estimation in High-Rise Buildings Based on Green’s Function

Abstract

High-rise buildings are inherently vulnerable to substantial wind-induced forces. The increasing complexity of building designs has posed challenges in calculating wind loads, while traditional methods involving physical models have proven to be intricate and time-consuming. In order to overcome these obstacles, this paper investigates a theoretical methodology aimed at streamlining the computation of wind loads. In the initial theoretical exploration, a simplified mathematical model based on Green’s function is introduced to take into account the interaction between wind loads and building geometry, while the model is not user-friendly and difficult to solve for complex polygonal buildings. To overcome this challenge, the study incorporates numerical simulations to extend the ideas and refine the methodology. To simplify the problem from a three-dimensional to a two-dimensional context, a bold tangential field assumption is made, assuming the wind pressure distribution remains similar across horizontal sections at different heights. The Schwarz–Christoffel formulation is then employed to facilitate the transformation. By integrating Green’s functions and conformal mapping to solve potential flow problems beyond the boundary layer, a comprehensive mathematical derivation is established. The above broadens the applicability of the mathematical theory and provides a new direction for estimations of high-speed wind load on buildings.

1. Introduction

To ensure structural safety, wind load design has become an indispensable aspect of high-rise building design. As building materials and technologies continue to advance, high-rise buildings exhibit characteristics of lightweight construction, reduced stiffness, and low damping, often with unique architectural designs. Therefore, accurate wind load calculations are quite important throughout the processes of site selection, design, construction, monitoring, and maintenance of high-rise structures.

In reviewing the historical development of wind load analysis, it is evident that early methods relied heavily on simplified static calculations and empirical data [1]. As computational capabilities advanced, dynamic simulations, such as those provided by computational fluid dynamics (CFD), became more prevalent [2]. These simulations offer a more detailed understanding of complex fluid–structure interactions, especially for buildings with non-traditional geometries [3].

Traditional methods, such as the static force equation approach derived from fundamental principles of fluid mechanics, have been supplemented by dynamic simulations that account for the transient nature of wind and its effects on structures [4]. The comparison of these methods reveals that while traditional approaches provide quick estimates, they often lack the precision required for designing safer, more efficient structures in diverse urban environments [5].

Furthermore, the use of wind tunnel testing has evolved from simple force measurements to sophisticated aeroelastic testing that can simulate real-world conditions more accurately [6]. This method, despite its higher cost and complexity, provides crucial validation for computational models, ensuring that both theoretical and empirical approaches align with real-world behaviors [7]. As a result, the investigation of wind loads on high-rise buildings, contingent upon the building structure’s shape, often necessitates intricate theoretical derivations [8], time-intensive numerical simulations, costly wind tunnel testing, and field measurements.

Theoretical investigations during the early stages primarily focused on establishing the theoretical framework of fluid mechanics, which scholars employed to address various flow-related problems. In fluid dynamics, the equations that describe the motion of fluids typically take the form of partial differential equations, such as the Navier–Stokes equations. Analytically solving these equations for complex geometries and boundary conditions is often challenging. However, Green’s functions provide an approach to address these difficulties by breaking down the problem into simpler components. It illustrates how the fluid responds to a localized input, such as a point force or a localized concentration of mass or energy. Green’s function encapsulates the influence of the source on the fluid flow and mathematically describes the resulting velocity, pressure, or other flow properties [9]. As early as the 1950s, John [10] systematically elaborated and popularized Green’s function in the general linear differential equation. And he solved the motion of a floating body in waves whose motion is simply harmonic by Green’s function method [11]. Pozrikidis [12] examined the physical transformations of the flow involved in obtaining singly and doubly periodic Green’s functions from their triply periodic counterparts. Marshall et al. [13] proposed a fast method to characterize wind fields by utilizing the free-space Green’s function. This approach offers an alternative to complex physics-based models or other surrogate models that necessitate volumetric discretizations. It is worth noting that mathematical theory is infrequently employed in the study of wind fields, with the majority of research conducted using numerical simulations or model experiments [14,15].

With the advent and rapid advancement of computer technology, researchers have increasingly turned to computational fluid dynamics (CFD) for simulating wind effects on buildings. For instance, Chay et al. [16] employed CFD and mathematical methods to investigate wind loads on buildings during the downburst, a specific meteorological phenomenon. They conducted an analysis of existing numerical models applied to the downburst and evaluated their strengths and weaknesses. Another notable study by Hur et al. [17] involved the comparison of wind load simulations using CFD software STAR-CD with experimental results obtained from wind tunnel tests at four stations in Korea. The comparison indirectly demonstrated the success of CFD simulations in practical scenarios. Furthermore, Lu et al. [18] focused on the study of wind loads on long-span complex roof structures using the Large Eddy Simulation (LES) technique. Their research aimed to explore and optimize an efficient approach where numerical results could capture more detailed information compared to data from tests. Thordal, Marie Skytte et al. [19] assessed the accuracy of CFD simulations in predicting wind loads on high-rise buildings with modified corners without prior knowledge of wind tunnel results and found that the mean surface pressures and peak structural responses were in excellent agreement between CFD and experimental results.

Along with CFD, wind tunnel testing has developed rapidly over the last 20 years. Researchers have leveraged wind tunnel experiments to gain insights into wind field characteristics and wind pressure on high-rise buildings. For instance, Snabjörnsson [20] carried out field studies of wind field characteristics and wind pressure in a high-rise building in urban Iceland. The results obtained from wind tunnel tests were utilized to refine and validate the measured data. In a similar vein, Tamura et al. [21] conducted a series of wind tunnel tests on 31 different high-rise building models with varying cross-sections. The objective of their study was to analyze the aerodynamic characteristics exhibited by these models. Additionally, Matza Gusto Andika and Fariduzzaman [22] conducted wind tunnel testing to analyze vibration in high-rise buildings due to wind load, focusing on vortex-induced vibration and lock-in phenomenon, finding that vortex shedding at specific wind speeds can significantly amplify building vibrations. Qinhua Wang and Buwen Zhang [23] investigated the wind-induced responses and wind loads on a super high-rise building with various cross-sections and a high side ratio through wind tunnel tests, revealing that maximum wind-induced base overturning moments and acceleration responses occur at 60° or 330° wind directions, significantly influenced by the aerodynamic interference of surrounding buildings.

In recent years, on-site monitoring of wind loads on building structures has been increasingly used in engineering. Notably, Kijewski-Correa et al. [24,25] have developed a field measurement system to investigate the dynamic response of three high-rise buildings in Chicago while also conducting field measurements on a high-rise building in Boston and Seoul. Through their work, they have validated the reliability of prediction methods and underscored the necessity for comprehensive monitoring of tall buildings with varying structural forms in diverse wind environments in future studies. Furthermore, Kuok et al. [26] have conducted full-scale on-site monitoring of a 22-story reinforced concrete building during the landfall of Typhoon Vicente, focusing on investigating the structural performance of the building under extreme wind loads.

As demonstrated in the aforementioned studies, wind loads assume a critical role in various research domains. Efficient and accurate estimation of wind loads on buildings can lead to substantial reductions in calculation time and cost. Consequently, this paper is organized into six sections to provide a comprehensive exploration of the topic. Section 1 (the current section) introduces the subject matter and presents a summary of relevant studies. Section 2 elucidates the simplification of the Navier–Stokes equation by leveraging principles from fluid mechanics and mathematical physics. In Section 3, the expansion and inspiration garnered from numerical simulations in relation to theoretical derivations are discussed. Building upon a bold assumption of a tangential flow field, Section 4 outlines the utilization of Green’s function method and conformal mapping technique to address potential flow problems beyond the boundary layer. Section 5 focused on a specially shaped structure, utilizing mathematical analysis software alongside the SCtool for comprehensive computational evaluation. Finally, Section 6 presents the conclusions and offers recommendations for further research endeavors.

2. Preliminary Theoretical Research

2.1. Model Simplification in Theoretical Fluid Mechanics

In the loaded code for the design of building structures issued by the Ministry of Housing and Urban-Rural Development of the People’s Republic of China (MOHURD), the statistical sampling of local wind speed follows the Marshalle Gumbel distribution and is based on the annual maximum values. The return period of the basic wind speed is set to 50 years in accordance with the building code. As the wind travels from high altitude to ground level, its speed diminishes due to the presence of buildings. Based on the potential flow theory, it is expected that the flow field adheres to fundamental principles in fluid dynamics, including the continuity equation, the momentum equation, and the mathematical model governing boundary layer flow. Under a kind of special condition that the wind with basic wind speed acts on building uninterruptedly, the wind field reaches a theoretical steady state, namely,

$$\frac{\partial }{\partial t}=0.$$

(1)

According to the measured data of typhoons, it has been observed that the maximum wind velocities generally remain below 100 m/s. By considering the definition of the Mach number [27], which is the ratio of the maximum wind velocity to the speed of sound, it can be inferred that this ratio is typically below 0.3. Consequently, assuming a constant air density, the aerodynamic problem associated with wind loads on buildings can be treated as an incompressible problem.

The continuity equation in fluid dynamics is a mathematical statement of the conservation of mass. It asserts that mass cannot be created or destroyed within a flow; it can only be redistributed. This equation is essential for incompressible flow (where the fluid density is constant), which is commonly assumed in aerodynamic studies involving buildings and other structures.

Continuity equation,

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,$$

(2)

The momentum equations in fluid dynamics, often referred to as the Navier–Stokes equations, describe how the velocity field of a fluid evolves under the influence of various forces. These equations are derived from Newton’s second law of motion and account for viscous effects in the fluid.

Momentum equations:

x-direction,

$$\rho \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+\mu \Delta u,$$

(3)

y-direction,

$$\rho \left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)=-\frac{\partial p}{\partial y}+\mu \Delta v,$$

(4)

and z-direction,

$$\rho \left(u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu \Delta w,$$

(5)

where $\rho$ is the density of the fluid, p is pressure, and $\mu$ is the viscosity coefficient.

It should be noted that the analysis considers air as a Newtonian fluid and neglects the effect of mass forces. By assuming air as a Newtonian fluid, the study focuses on the dominant viscous effects and the behavior of fluid flow based on shear stress and velocity gradients.

Boundary layer theory deals with the behavior of fluid flow in the vicinity of a boundary (like the surface of a building). By the boundary layer equations,

$$\left\{\begin{array}{l}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,\\ \rho \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+\mu \frac{{\partial }^{2}u}{\partial z^2},\\ \rho \left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)=-\frac{\partial p}{\partial y}+\mu \frac{{\partial }^{2}v}{\partial z^2},\\ \frac{\partial p}{\partial z}=0,\end{array}\right.$$

(6)

According to boundary layer theory, the larger the Reynolds number of the fluid, the thinner the boundary layer. Estimate the thickness δ of the boundary layer roughly by the formula

$$\frac{\delta }{L}=O\left(\frac{1}{\sqrt{{\mathrm{Re}}_L}}\right),$$

(7)

A comparison of the order of magnitude shows that

$$\delta \sim \frac{L}{\sqrt{\mathrm{Re}}}=\frac{L}{\sqrt{\frac{LU}{\nu }}}=\sqrt{\frac{L\nu }{U}}$$

(8)

The coefficient of viscosity of the air is known to be v = 14.8 × 10⁻⁶ m²/s. Referring to the Beaufort scale, ranging from 20.7 to 61.2 m/s correspond to magnitudes from 8 to 17, respectively. In the case of high-rise buildings, their face width and depth typically do not exceed 100 m. As a result, the corresponding boundary layer thickness, denoted as δ, falls within the range of 0.0049–0.0085 m. When compared to the building’s characteristic length, L = 100 m, this thickness is considered negligible. Consequently, it can be assumed that the surface of the building coincides with the boundary layer surface.

The fundamental objective of fluid mechanics is to determine the forces and moments exerted by a fluid on an object. To achieve this, it is crucial to ascertain the pressure distribution surrounding the object. The Lagrangian integral can be employed to establish a relationship between the pressure and the velocity field. Specifically, a velocity potential exists, which can be expressed as follows.

$$\nabla \phi =\overrightarrow{V}=\left(\begin{array}{ccc}u,& v,& w\end{array}\right),$$

(9)

Then continuity equation can be transformed into the following Laplacian equation, yielding the following expression:

$$\nabla ·\nabla \phi =\Delta \phi =0.$$

(10)

This incompressible and irrotational problem is simplified to a mathematical physics problem. There is a wind field with a constant wind speed and a fixed direction,

$$\overrightarrow{W}=\left(\begin{array}{ccc}{w}_x,& w_y,& w_z\end{array}\right)$$

(11)

blowing construct Ω. For example, when constant wind,

$$\overrightarrow{W}=\left(1,0,0\right)$$

(12)

then, wind flow occurs along the x-axis. Theoretically, wind direction can be chosen arbitrarily. The points of the surface ∂Ω of the construct are affected by the wind, and their wind speeds are represented by $H\left(\begin{array}{ccc}x,& y,& z\end{array}\right)$. Then,

$$\left\{\begin{array}{cc}\Delta \phi =0,\hfill & in {ℝ}^3\backslash \mathsf{\Omega } ,\\ \frac{\partial \phi }{\partial \overrightarrow{n}}=H\left(\begin{array}{ccc}x,& y,& z\end{array}\right), \hfill & on \partial \mathsf{\Omega },\\ \underset{\left|x\right|\to \infty }{\lim }\phi =\overrightarrow{W}·\overrightarrow{x},\hfill & \end{array}\right.$$

(13)

$\frac{\partial \phi }{\partial \overrightarrow{n}}=\nabla \phi ·\overrightarrow{n}$ means that wind speed is perpendicular to surface ∂Ω. Simplifying the equations, let

$$\mathsf{\Phi }=\phi -\overrightarrow{W}·\overrightarrow{x},\tilde{H}\left(\begin{array}{ccc}x,& y,& z\end{array}\right)=H\left(\begin{array}{ccc}x,& y,& z\end{array}\right)-\overrightarrow{W}·\overrightarrow{n},$$

(14)

then,

$$\left\{\begin{array}{cc}\Delta \mathsf{\Phi }=0,\hfill & in {ℝ}^3\backslash \mathsf{\Omega }, \\ \frac{\partial \mathsf{\Phi }}{\partial \overrightarrow{n}}=\tilde{H}\left(\begin{array}{ccc}x,& y,& z\end{array}\right), \hfill & on \partial \mathsf{\Omega },\\ \underset{\left|x\right|\to \infty }{\lim }\mathsf{\Phi }=0.\hfill & \end{array}\right.$$

(15)

In mathematics, Equation (16) is called Laplace’s equation with the Neumann boundary condition in the exterior area. If the aforementioned equations can be solved, the velocity gradient at each point in the exterior of the structure can be determined. Moreover, the pressure can be obtained using the momentum equations. Therefore, the problem can be solved.

2.2. Solving Equations in Mathematical Physics

Theorem 1

(Representation formula using Green’s function [28]). If $\mathsf{\Phi }\in C^2\left({ℝ}^3\backslash \mathsf{\Omega }\right)$ solves Equation (16), then

$$\mathsf{\Phi }\left(x\right)=-{\displaystyle {\int }_{\partial \mathsf{\Omega }}H\left(y\right)}G\left(x,y\right)dy+C,$$

(16)

where $x\in {ℝ}^3\backslash \mathsf{\Omega }$, $y\in \partial \mathsf{\Omega }$, and $G\left(x,y\right)$ are called Green’s function, which satisfies the following equation:

$$\left\{\begin{array}{cc}\Delta G=\delta \left(x-x_0,y-y_0,z-z_0\right), \hfill & in {ℝ}^3\backslash \mathsf{\Omega }, \\ \frac{\partial G}{\partial \overrightarrow{n}}=-\frac{1}{\left|\partial \mathsf{\Omega }\right|}, \hfill & on \partial \mathsf{\Omega },\\ \underset{\left|x\right|\to \infty }{\lim }G=0.\hfill & \end{array}\right.$$

(17)

A formula for the solution of the Neumann problem, Equation (17), is given by Theorem 1, provided Green’s function G for a given domain $\mathsf{\Omega }$ can be constructed. Whether it can be done or not quite depends on the shape of the domain. As is all known, Equation (18) with a spherical domain $\mathsf{\Omega }$ whose radius is R has solutions in many books on Geoscience [29]. However, because Laplace’s fundamental solutions are radially symmetric, it is not possible to solve the general solution of the Green–Neumann exterior equation for any region uniformly when the domain is not a ball. There are too many difficulties to tackle, even if the domain is rectangular.

Considerable time and effort were dedicated to exploring different mathematical methods for solving the “rectangular domain”. However, no significant progress was achieved. Consequently, alternative approaches were pursued to explore new avenues and provide inspiration.

3. Further Theoretical Derivation

Building upon the aforementioned research, a novel proposition is put forth. The concept revolves around the idea of dividing the flow field akin to cutting a woodblock. By segmenting the boundary conditions into sections, it becomes possible to reduce the problem from a three-dimensional (3-D) scenario to a two-dimensional (2-D) one.

Most high-rise buildings are polygonal and equal cross-section structures like Figure 1. If there is an assumption that the pressure distribution of the horizontal sections at different heights of the study subject is close to the same (i.e., the height has only a weak effect on the wind pressure on high-rise buildings) and the problem is translated into the research of wind field in the exterior area of the contour $\partial \Gamma$ of cross-section $\Gamma$ (level $z=z_0$) with wind speed on the contour $\partial \Gamma$ $\tilde{H}\left(x,y,z_0\right)$, then

$$\left\{\begin{array}{cc}\Delta G=\delta \left(x-x_0,y-y_0\right), \hfill & in {ℝ}^2\backslash \Gamma , \\ \frac{\partial G}{\partial \overrightarrow{n}}=-\frac{1}{\left|\partial \Gamma \right|}, \hfill & on \partial \Gamma ,\\ \underset{\left|x\right|\to \infty }{\lim }G=0.\hfill & \end{array}\right.$$

(18)

As stated above, it is difficult to solve Green’s function with an arbitrary domain in 3-D. However, many mathematical skills take effect with the descendent of dimension. Conformal mapping, which is the most useful and powerful tool in many fields like pattern recognition, can be used. The characteristics of conformal mapping are as follows:

• The mapping is a univalent analytic function;
• The angle between curves after mapping is the same as that before mapping;
• The real part of and the imaginary part of mapping is the harmonic function.

Theorem 2

(Schwarz–Christoffel formula [30]). Let $\omega =f\left(z\right)$ be a conformal mapping from the upper half-plane $\mathrm{Im} \mathrm{z} > 0$ to the exterior of polygonal P, whose vertex angles are

$${\beta }_k\pi \left(0<{\beta }_k<2,k=1,2,3,\dots ,n,{\displaystyle \sum _k{\beta }_k=n+2}\right)$$

(19)

and the points of the real axis $x_k\left(-\infty <x_1<\dots <x_n<+\infty \right)$ correspond to the vertex points of the polygonal, then

$$\omega =C{\displaystyle \underset{0}{\overset{z}{\int }}\frac{{\left(t-x_1\right)}^{{\beta }_1-1}\dots {\left(t-x_n\right)}^{{\beta }_n-1}}{\left(t-\alpha \right)\left(t-\overline{\alpha }\right)}}dt+C_1$$

(20)

where $C,C_1$ is constant and  $\alpha \left(\mathrm{Im}\alpha >0\right)$ corresponds to the infinity point of the upper plane.

Because the contour $\partial \Gamma$ of the building is commonly a simple polygon, Theorem 2 is useful. Firstly, ${ℝ}^2\backslash \Gamma$ ($\zeta =x+yi$) can be transformed into the upper half-plane by the inverse mapping of Theorem 2. Secondly, the upper half-plane can be transformed into the interior of the unit circle by other conformal mappings like $\frac{\omega -i}{\omega +i}$. Finally, the interior of the unit circle transforms into the exterior of the unit circle by $\frac{1}{\omega }$. Thus, the shape of the cross-section $\Gamma$ turned into the unit circular $\mathrm{O}$ by a series of conformal mappings, which is denoted by $\eta \left(\xi \right)$, and Equation (19) translates into the following:

$$\left\{\begin{array}{cc}\Delta K=\delta \left(\xi -{\xi }_0\right)\hfill & in {ℝ}^2\backslash \mathrm{O} \\ \frac{\partial K}{\partial r}=-\frac{1}{2\pi }\hfill & on \partial \mathrm{O}\\ \underset{\left|x\right|\to \infty }{\lim }K=0\hfill & \end{array}\right.$$

(21)

If Equation (22) can be worked out, then Equation (19) can be represented by $G\left(\eta \left(\xi \right)\right)$. Consequently, Equation (16) can be figured out.

Now, to solve Equation (16) with the nonzero boundary condition $\tilde{h}\left(x,y\right)$, it is shown as follows:

$$\left\{\begin{array}{cc}\Delta K=0\hfill & in {ℝ}^2\backslash \mathrm{O} \\ \frac{\partial K}{\partial r}=\tilde{h}\left(x,y\right)\hfill & on \partial \mathrm{O}\\ \underset{\left|x\right|\to \infty }{\lim }K=0\hfill & \end{array}\right.$$

(22)

Then, by Theorem 1, the solution of Equation (23) can be represented by

$$\begin{array}{ll}K\left(x_0,y_0\right)& =K\left(r_0\cos {\alpha }_0,r_0\sin {\alpha }_0\right)\\ & ={\displaystyle {\int }_{\partial \mathsf{\Omega }}G\left(x_0,y_0,x,y\right)\tilde{h}\left(x,y\right)dxdy}\\ & ={\displaystyle {\int }_{\left|\mathrm{r}\right|=1}G\left(r_0,{\alpha }_0,\alpha \right)}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)d\alpha \end{array}$$

(23)

and a separation of variables can be used in Equation (23).

$$\Delta K=0\Rightarrow \frac{{\partial }^{2}K}{\partial r^2}+\frac{1}{r}\frac{\partial K}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}K}{\partial {\alpha }^2}=0.$$

(24)

Then, by $\underset{\left|x\right|\to \infty }{\lim }K=0$,

$$K\left(r_0\cos {\alpha }_0,r_0\sin {\alpha }_0\right)={{\displaystyle \sum _{n=0}^{\infty }\left(\frac{1}{r_0}\right)}}^n\left(A_n\cos n{\alpha }_0+B_n\sin n{\alpha }_0\right)$$

(25)

$${\left.\frac{\partial }{\partial r}K\left(r\cos \alpha ,r\sin \alpha \right)\right|}_{r=1}={\displaystyle \sum _{n=0}^{\infty }\left(A_n\cos n\alpha +B_n\sin n\alpha \right)\left(-n\right)}$$

(26)

and by Fourier series analysis and Equation (27),

$$\left(\begin{array}{c}{A}_n\\ B_n\end{array}\right)=-\frac{1}{n\pi }{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}}\left(\cos \alpha ,\sin \alpha \right)\left(\begin{array}{c}\cos n\alpha \\ \sin n\alpha \end{array}\right)d\alpha$$

(27)

Then,

$$\begin{array}{ll}K\left(r_0\cos {\alpha }_0,r_0\sin {\alpha }_0\right)& ={{\displaystyle \sum _{n=0}^{\infty }\left(\frac{1}{r_0}\right)}}^n\left(-\frac{1}{n\pi }\right){\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\left(\cos {\alpha }_0\cos \alpha +\sin {\alpha }_0\sin \alpha \right)d\alpha \\ & =\left(-\frac{1}{\pi }\right){{\displaystyle \sum _{n=0}^{\infty }\left(\frac{1}{r_0}\right)}}^n\frac{1}{n}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\cos \left({\alpha }_0-\alpha \right)d\alpha \\ & =\left(\frac{1}{\pi }\right){{\displaystyle \sum _{n=0}^{\infty }\left(\frac{1}{r_0}\right)}}^n\frac{1}{n}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\frac{-e^{in\left({\alpha }_0-\alpha \right)}-e^{-in\left({\alpha }_0-\alpha \right)}}{2}d\alpha \\ & =\left(\frac{1}{2\pi }\right)\left\{{{\displaystyle \sum _{n=0}^{\infty }\left(\frac{1}{r_0}\right)}}^n\frac{1}{n}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\left(-e^{in\left({\alpha }_0-\alpha \right)}\right)d\alpha \right.\\ & \left.+{{\displaystyle \sum _{n=0}^{\infty }\left(\frac{1}{r_0}\right)}}^n\frac{1}{n}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\left(-e^{-in\left({\alpha }_0-\alpha \right)}\right)d\alpha \right\}\\ & =\left(\frac{1}{2\pi }\right)\left\{{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}{{\displaystyle \sum _{n=0}^{\infty }\left(\frac{1}{r_0}\right)}}^n\frac{1}{n}\left(-e^{in\left({\alpha }_0-\alpha \right)}\right)d\alpha \right.\\ & \left.+{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}{{\displaystyle \sum _{n=0}^{\infty }\left(\frac{1}{r_0}\right)}}^n\frac{1}{n}\left(-e^{-in\left({\alpha }_0-\alpha \right)}\right)d\alpha \right\}\\ & =\left(\frac{1}{2\pi }\right)\left\{{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\log \left(1-\frac{1}{r_0}{e}^{i\left({\alpha }_0-\alpha \right)}\right)d\alpha \right.\\ & \left.+{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\log \left(1-\frac{1}{r_0}{e}^{-i\left({\alpha }_0-\alpha \right)}\right)\right\}\\ & =\left(\frac{1}{2\pi }\right){\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\log \left(1-\frac{1}{r_0}{e}^{i\left({\alpha }_0-\alpha \right)}\right)\left(1-\frac{1}{r_0}{e}^{-i\left({\alpha }_0-\alpha \right)}\right)d\alpha \\ & =\left(\frac{1}{2\pi }\right){\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\log \frac{\left(r_0-e^{i\left({\alpha }_0-\alpha \right)}\right)\left(r_0-e^{-i\left({\alpha }_0-\alpha \right)}\right)}{r_0{ }^2}d\alpha \\ & =\left(\frac{1}{2\pi }\right){\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\tilde{h}\left(\cos \alpha ,\sin \alpha \right)}\log \frac{r_0^2+1-2r_0\cos \left({\alpha }_0-\alpha \right)}{r_0^2}d\alpha \end{array}$$

(28)

Because of the Strong Maximum Principle [28], the solution of Laplace’s equation is unique and $h\not\equiv 0$, then Green’s function is

$$G\left(r_0,{\alpha }_0,\alpha \right)=\frac{1}{2\pi }\log \frac{r_0^2+1-2r_0\cos \left({\alpha }_0-\alpha \right)}{r_0^2}$$

(29)

Then, using $\eta \left(\xi \right)$, the Green’s function of Equation (19) with an arbitrary polygon is the following.

$$G\left(x_0,y_0,x,y\right)=\frac{1}{2\pi }\log \frac{{\left|\eta \left({\xi }_0\right)\right|}^2+1-2\mathrm{Im}\eta \left({\xi }_0\right)\mathrm{Im}\eta \left(\xi \right)-2\mathrm{Re}\eta \left({\xi }_0\right)\mathrm{Re}\eta \left(\xi \right)}{{\left|\eta \left({\xi }_0\right)\right|}^2}$$

(30)

where $\xi =x+yi$ and ${\xi }_0=x_0+y_{0}i$.

Finally, Equation (23) can be solved, and the solution is the following:

$$\mathsf{\Phi }\left(x_0,y_0\right)={\displaystyle {\int }_{\partial \Gamma }\tilde{H}\left(x,y\right)}G\left(x_0,y_0,x,y\right)dxdy$$

(31)

where $\left(x_0,y_0\right)\in {ℝ}^2\backslash \mathsf{\Omega }$.

The potential function of the surface of the building is solved as the following:

$$\phi \left(x_0,y_0,z_0\right)=\mathsf{\Phi }\left(x_0,y_0,z_0\right)+\overrightarrow{W}·\overrightarrow{x}$$

(32)

where ${\overrightarrow{x}}_0=\left(x_0,y_0\right)$.

The gradient of pressure can be obtained by the momentum equation. Moreover, by a basic wind pressure formula

$$P_0=\left(P_{0x},P_{0y}\right)=\frac{1}{2}\rho \left(W_x^2,W_y^2\right)$$

(33)

The pressure of infinity is given. The pressure $P=\left(P_x,P_y\right)$ of the building can be estimated by

$$P_x={\displaystyle \underset{-\infty }{\overset{x}{\int }}\frac{\partial P}{\partial t}}dt+P_{0x}$$

(34)

Similarly, the pressure of the y-direction can be estimated.

4. Exploration by CFD

In the process of scientific research, numerical simulation is a powerful tool that can simulate and predict physical phenomena or system behaviors. It aids researchers in better understanding the behavior and mechanisms of complex systems and provides new ideas and methods for solving practical problems, such as revealing natural laws, optimizing engineering designs, and developing new materials.

The dimensional reduction assumption in the method of this paper is inspired by the following numerical simulation.

4.1. Model

An illustrative example is provided by a hypothetical high-rise building with a height of 47 m, intended for a nearshore location. To study the wind characteristics surrounding the structure, a numerical model utilizing the Reynolds Average Navier–Stokes (RANS) equations, in conjunction with a turbulence model, is employed. Figure 2 presents three visual perspectives of the simulated building.

The wind field is simulated using a widely used computational fluid dynamics (CFD) solver, considering the steady-state pressure distribution under a constant wind velocity of 38.78 m/s. The computational domain spans a size of 260 m × 160 m × 100 m, and the grid is refined using a nesting and layer-by-layer approach, resulting in three layers of refinement. The maximum mesh size is set to 4 m, while the minimum mesh size is 0.125 m. The mesh interval is further refined to 120 m × 60 m × 60 m. The air density is assumed to be 1.25 kg/m³, and the dynamic viscous coefficient is 1.8 × 10⁻⁵ Pa·s. The mesh structure and its refinement interval are depicted in Figure 3 and Figure 4, respectively.

The inlet boundary conditions are set as a constant velocity of 30 m/s, while the outlet boundary conditions allow for free-flowing conditions. The boundaries of the building’s walls and the ground surface are defined as non-slip boundary conditions. The finite volume method is employed to discretize Reynold’s equation. For gradient calculation, the pressure Poisson equation is chosen as second-order upwind. Turbulent kinetic energy and divergent term of energy dissipation employed Gaussian linear scheme. The relaxation factors for the numerical scheme have been carefully selected based on existing experience: 0.3 for the pressure term, 0.7 for the velocity term, 0.7 for the turbulent kinetic energy, and 0.7 for the dissipated energy. The remaining factors are set to 1.

Two grids with different levels of accuracy were employed to assess convergence. The first grid, denoted as mesh1, had a grid size ranging from 0.125 to 4.0 m, while the second grid, mesh2, had a finer resolution with a grid size ranging from 0.0625 to 2.0 m. Figure 5 illustrates a comparison of the combined wind load forces obtained from the two mesh simulations. After reaching a stabilized state, the wind load combined force of mesh1 was determined to be 1132.2 kN, whereas that of mesh2 was 1150 kN, resulting in a discrepancy of 1.5% between the two. It was evident that mesh1 fulfilled the convergence criteria and offered a balance between grid resolution and computational cost for conducting the simulation in this example.

4.2. Numerical Simulation Results

The velocity distribution around the building is shown in Figure 6, and the pressure distribution of the wind load exerted on the surface of the building is shown in Figure 7. Observing the figures, it is apparent that the wind pressure on the windward and leeward sides of the building exhibits a relatively uniform pattern, with minimal variation in relation to height. Furthermore, a noticeable disparity in wind pressure is observed between the shape of the building and its roof.

Seven lines of measurement at different heights are selected on the windward side, and the magnitude of the wind pressure on the lines of measurement is depicted, as shown in Figure 8. It can be seen that the trend of the wind pressure distribution exhibits a remarkable similarity. This indicates that the pressure distribution within the horizontal cross-section of the studied object remains largely consistent across different heights.

The aforementioned findings have provided new inspirations. By investigating Design Codes [31,32], it is found that the height parameters and variables of wind loads are not complicated for high-rise buildings, as they vary with height. However, with changes in the shape of the building, such as in cross-section, the derivation of the shape parameters is more complicated, and the deviation of the calculated results is larger.

In building design, the shape of a building frequently undergoes changes driven by factors such as customer requirements and construction needs. However, these shape modifications necessitate significant recalculations, leading to substantial rework and time investments, particularly in the realm of numerical simulations and experimental tests. Therefore, further mathematical solutions might offer a rapid resolution to such problems.

5. Example

For buildings with complex exterior structures, we refine the wind load calculation by employing a two-step simplification process: dimensional reduction and conformal mapping. The initial phase entails the diminution of the model from a tri-dimensional spatial framework to a bi-dimensional plane. Thereafter, conformal mapping is applied to transmute the complex morphology into a geometrically simplified form. This methodology enhances the tractability of wind load analysis on structures of complex geometry, rendering the computational model more amenable.

To underscore the adaptability of the proposed calculation methodology to structures of specific geometries, select the example from Section 4 as the subject of study, using mathematical analysis software and the SCtool tool. With the origin as the center, determine the vertex positions according to the structural contour.

This study is conducted over a discretized grid with an incremental step of 0.1 within the defined square region beyond the complex plane W1 at xarea = [−20, 20] and yarea = [−20, 20]. Utilizing the SCtool toolbox, mapping results are derived, as demonstrated in the corresponding Figure 9.

To refine the analysis of fluid dynamics, the model is abstracted through the exclusion of external forces, air viscosity, and compressibility. This simplification allows us to concentrate on the fundamental interactions between wind loads and structural surfaces, sidestepping the complexity of a fully dynamic airflow model while still providing insightful results.

Employing the theoretical framework of the Green–Neumann problem within a two-dimensional context, as delineated in Section 3, the calculation of the pressure distribution within the flow field is executed, with the findings presented in Figure 10.

Next, a comparative analysis of the calculation results with standards [32] and CFD results is conducted. The average load values on the windward side, according to standard formulas and CFD results, are compared with the example results, as shown in

Table 1. Comparison of wind load standards with CFD results.

Standard Calculation Results (N/m2)	CFD Simulation Segment Average Value (N/m2)	Difference
571.5	615.3	7.66%

and

Table 2. Comparison of wind load standards with example calculation results.

Standard Calculation Results (N/m2)	Example Results (N/m2)	Difference
571.5	637.9	11.62%

.

It is apparent that there is a certain difference between the standard values and numerical simulation results, with CFD results showing about an 8% deviation and the example results showing about a 12% deviation. The pressure distribution in the windward side of the example is higher in the middle and lower at the ends, generally consistent with the distribution trend of the CFD. The resultant wind load on the building, the standard value is 1134.1 kN, and the example result is 1132.2 kN, which is almost equal. Overall, the example results and distributions are greater than the numerical simulation results, which are greater than the standard value results, aligning well with standards and numerical simulations and conforming to practical engineering conditions.

6. Conclusions

This paper presents a theoretical approach aimed at simplifying the solution of wind loads. It differs from the conventional practice of extrapolating existing theories to specific applications. Instead, this study begins with fundamental theory and, when encountering obstacles, incorporates ideas derived from numerical simulations. Under the premise of satisfying engineering assumptions, a complete solution formula is obtained from a mathematical perspective, thereby expanding the applicability of the mathematical theory.

It can be summarized as follows:

(1)

From the numerical simulation results, it can be concluded that the wind pressure distribution trends in the horizontal sections of the building at various heights exhibit remarkable similarities.

(2)

Based on the assumption that the wind pressure distribution remains relatively consistent across different height levels, the research derived formulae for addressing the potential flow problem outside the boundary layer using Green’s function method and conformal mapping.

(3)

The application of dimension-reduced mathematical methods provides a reference for studying flow field and fluid dynamics in various engineering contexts. It involves the utilization of techniques that simplify the problem space, allowing for a more efficient and effective approach to tackling complex flow field problems. By simplifying the problem space, this method tackles complex flow field problems in a more streamlined and effective manner.

(4)

The flexible application of the Schwarz–Christoffel formula can provide some references for other research fields related to fluid dynamics.

(5)

When theoretical research encounters obstacles, the utilization of numerical simulation to find a breakthrough point may open up new ideas for numerous engineering problems.

(6)

The mathematical formulae developed in this study can be implemented in various computational tools to enhance efficiency. This facilitates architects and engineers in obtaining wind load data rapidly during the preliminary design stage.

This study emphasizes the crucial role of theoretical derivation in understanding the impact of high-speed wind loads on high-rise buildings. Estimations made during the preliminary design process can quickly and effectively support adjustments to the shape and structural reinforcement of areas heavily affected by wind. Because calculation deviations are greater than numerical simulation results, it is still necessary to refer to standards, numerical simulations, wind tunnel tests, and other methods during detailed local design to ensure the structural safety of buildings. Future research should focus on themes such as efficiency, accuracy, and the application of different computational methods.