A CFD Study on Wind Pressure Characteristics and Vortex-Induced Vibration of the Yingxian Wooden Pagoda

Abstract

The Yingxian Wooden Pagoda, a structure with a history spanning a thousand years, currently faces significant wind-induced safety risks. To understand the aerodynamic mechanism behind this issue, this study uses Computational Fluid Dynamics (CFD) with the Realizable k-ε turbulence model to perform high-fidelity transient simulations at wind speeds from 10 to 30 m per second. The results show that the highest positive pressure occurs on the sides of the windward face, while a large low-pressure vortex zone forms on the leeward side. The simulations include both the Kármán vortex street and the measurement of three-dimensional vortex-induced forces, marking a major advancement. A key finding is the synchronized period (ratio ≈ 1) of the along-wind and cross-wind forces, which differs from streamlined cylinders and is due to the pagoda’s unique octagonal shape. The force amplitudes increase exponentially with wind speed, while the average drag and lift have a quadratic relationship. Additionally, a new shape-specific correction factor of 0.875 is introduced to adjust the classical Strouhal formula, which greatly improves prediction accuracy for this type of ancient structure. This study offers both a theoretical foundation and a practical “digital wind tunnel” method for assessing wind-induced risks and supporting the safety monitoring of historic timber structures.

1. Introduction

The Yingxian Wooden Pagoda (Figure 1, left), located in Shanxi Province, China, is one of the oldest surviving multi-story timber structures in the world. Officially called the Sakyamuni Pagoda of the Fogong Temple, this pavilion-style architecture was built in 1056 AD during the Liao Dynasty [1]. The pagoda stands at a total height of 65.86 m and has a base diameter of 30.27 m, featuring an octagonal plan in all its stories. It consists of nine structural levels: five visible (ming) and four concealed (an), supported by a 4-m-high stone platform [2]. Externally, the pagoda features five main tiers with six stories: the first tier has a double-eave roof, while the upper four tiers each have a single eave. The pagoda’s notable structural strength is due to its unique “outer and inner groove” (waicao and neicao) column system used in both the visible and hidden stories. In the visible stories, the inner groove is defined by eight core columns, creating a spacious chamber for Buddhist statues. In contrast, additional pin columns provide stability within the outer groove. The hidden stories act as vital structural diaphragms, incorporating radial and circumferential bracing to create a rigid ring-beam effect, as shown in Figure 1 (right) [3]. The numerical simulations and mesh generation were performed using the open-source software OpenFOAM (v2312).

The Yingxian Wooden Pagoda is an outstanding example of ancient Chinese timber architecture. However, its structure has suffered significant damage over nearly a thousand years due to environmental wear, material aging, seismic activity, wind forces, and human impact. These accumulated damages have weakened both the static and dynamic stability of the pagoda, raising serious safety concerns. Visible issues include tilting, twisting of the upper levels, and localized damage to timber components. In particular, repeated wind loads have accelerated material deterioration, greatly increasing the structure’s vulnerability during strong winds or earthquakes. Consequently, a thorough evaluation of the pagoda’s current safety performance and its response to environmental forces (especially wind) has become a crucial part of conservation efforts. To address this, a joint research team from the China Cultural Heritage Research Institute and Beijing Jiaotong University carried out a full-scale field measurement in 2011 to assess wind loads and structural responses. The study measured wind conditions around the structure, the surface pressure distribution induced by the wind, and the associated structural behavior. Findings showed substantial distortions both globally and locally, especially a gradual inward tilt of the southwest corner column of the second story. The analysis linked these deformations to wind-induced thrust forces mainly coming from the southwest, which matches the prevailing wind pattern [4]. These results confirmed that wind load significantly contributes to the current damage, highlighting the need for a more detailed study of how wind interacts with the structure.

Current methods for analyzing structural wind pressure distributions include field measurements, wind tunnel tests, and Computational Fluid Dynamics (CFD) simulations, each with its own advantages and limitations. Field measurements provide direct insights into real wind conditions but are limited by high costs and interference from structural deformation, making long-term studies difficult [4]. Although wind tunnel tests can mimic flow characteristics using scaled models with well-controlled variables, they are often constrained by long durations and substantial costs [5,6]. In contrast, CFD simulations use computational models to solve the governing equations of fluid dynamics, offering significant cost savings, improved efficiency, and greater accuracy. This approach is especially useful for structures like ancient buildings, as it avoids the impracticality of repeated physical experiments [7,8]. Recent research confirms that wind load is a key factor in the current damage and ongoing tilt of the Yingxian Wooden Pagoda [9]. For instance, field data directly link the pagoda’s notable northeast tilt to the persistent action of prevailing northwest winds, with quantified inter-story drifts in critical layers providing concrete evidence of wind-induced deformation [10]. To understand the aerodynamic mechanisms behind this, such as unsteady vortex shedding and three-dimensional flow interactions that influence the structural response, a high-fidelity transient CFD simulation acts as an essential digital wind tunnel. It offers unmatched flexibility and scalability for detailed analysis, although results depend on turbulence model and boundary condition choices.

Under specific inflow conditions, the steady flow around a bluff body will periodically generate and shed vortices alternately from both sides. The resulting pulsating loads can induce forced vibrations in the structure, so it is important to consider the Kármán vortex street effect in wind-impact simulations of wooden Pagodas. When wind moves over a cylindrical or prismatic body, the rear part enters a flow deceleration and pressure recovery zone, leading to boundary layer separation. This separation pattern changes with the Reynolds number (Re). Once Re > 90, the shear layers separate and roll up to form discrete vortices that are shed alternately, creating a regular vortex pattern in the wake—the Kármán vortex street. This street induces periodic variations in surface pressure and shear stress, which can excite structural vibrations in both the wind and cross-wind directions, with the fundamental frequency matching the vortex shedding frequency. If this frequency matches the structure’s natural frequency, resonance can occur, potentially causing large vibrations [11]. To explore this phenomenon, Liu Yu et al. [11] used computational simulations to study the Kármán vortex shedding and wake patterns of a square column at different Reynolds numbers. They reported their force coefficients and Strouhal numbers.

Based on existing research, this paper analyzes the flow behavior around the Yingxian Wooden Pagoda under different wind speeds. The shedding patterns of Karman vortices and the trailing vortex characteristics of this tower, which has a distinctive octagonal shape and significant height, are specifically examined. The study establishes the relationships between wind speed, wind pressure distribution, and vortex-induced vibration patterns, and calculates the force coefficient and Strouhal number of the Yingxian Wooden Pagoda.

2. Computational Process

Computational Fluid Dynamics (CFD) is a vital tool for simulating and analyzing complex fluid flow phenomena by numerically solving the Navier–Stokes equations. A typical CFD workflow includes three main stages: pre-processing, which involves geometry modeling, mesh generation, and setting boundary conditions; solving, where the discretized governing equations are iteratively solved to determine the flow field variables; and post-processing, which focuses on visualizing and quantitatively analyzing the results [12,13]. The Finite Volume Method (FVM) [14], commonly used in commercial and research CFD codes, is particularly well-suited for this task. In the FVM, conservation laws are discretely enforced over control volumes defined by the computational grid. This method inherently guarantees the conservation of mass, momentum, and energy, and provides considerable flexibility for managing complex geometries with both structured and unstructured meshes.

2.1. Geometric Models, Fluid Domains, and Boundary Conditions

Preprocessing is the most time-consuming and tedious part of the CFD simulation process [15]. The selection of the computational domain, mesh generation, and the number and quality of the mesh elements all significantly influence the accuracy of the simulation results. The geometric model was created from drawings provided by the Shanxi Provincial Institute of Ancient Architecture Conservation (as shown in Figure 2). To improve computational accuracy and efficiency, the wooden pagoda was simplified by omitting certain details while keeping its key features. This simplification approach is supported by a recent comparative study on the same pagoda, which showed that while more geometric detail enhances local accuracy, a model that maintains the main octagonal shape and essential protruding elements (such as railings and eaves) can effectively represent the main flow mechanisms and overall wind pressure patterns needed for vortex-induced vibration analysis [16]. This method greatly increased computational efficiency without sacrificing accuracy.

Based on established practices in computational wind engineering [17], the external flow field was modeled as a rectangular domain. The dimensions of the computational domain relative to the pagoda height (H) were defined as follows: 12H in the crosswind direction, 4H in the along-wind direction, and 3H in the vertical direction. The pagoda was positioned approximately one-third of the domain length from the inlet to ensure adequate upstream flow development and a sufficiently long downstream region for wake formation, thereby preventing spurious flow reversals at the outlet [17]. The resulting blockage ratio, defined as the ratio of the projected frontal area of the pagoda to the cross-sectional area of the computational domain, was maintained at 2.5%. This value is low enough to minimize the influence of the lateral boundary conditions on the flow around the structure.

2.2. Turbulence Models and Settings

Numerical simulation of turbulent flows requires choosing an appropriate turbulence model. The Reynolds-Averaged Navier–Stokes (RANS) approach and Large Eddy Simulation (LES) are the two main methods used for this purpose. The RANS framework, commonly used in engineering because of its computational efficiency, relies on the Reynolds decomposition of flow variables into time-averaged and fluctuating components. This process converts the transient Navier–Stokes equations into steady-state forms. Conversely, LES directly resolves the large, energy-carrying eddies while modeling the effects of smaller, universal scales with a subgrid-scale model. However, LES’s high computational cost, long simulation times, and strict grid-resolution requirements make it less suitable for many engineering-scale problems. Recent studies on traditional timber structures support this, showing that the steady-state RANS approach is a practical and effective method for wind-induced vibration analysis [18]. Considering these factors, the RANS approach was chosen for this study. Specifically, the Realizable k-ε model [19] was used, as previous research has shown its effectiveness in accurately predicting wind pressure distribution on the windward, side, and leeward surfaces of wooden tower structures [17]. This model includes a vorticity-viscosity coefficient,

$$C_{\mu }={\displaystyle \frac{1}{A_0+A_s\frac{Uk}{\epsilon }}}$$

(1)

The model accounts for the strain rate, characteristic speed, turbulent kinetic energy, and turbulent dissipation rate. This method can accurately predict the location of the separation point around a cylinder and the pressure distribution [15]. Additionally, the Realizable k-ε model shows good robustness in high-Reynolds-number flows [20]. Therefore, the SIMPLE algorithm is chosen for flow-field simulation in this paper, which was executed using the open-source CFD toolbox OpenFOAM (v2312). Moreover, during CFD simulations, convergence criteria must be set to ensure reliable and accurate results. The convergence criteria used in this study include setting up a velocity monitoring line behind the building to check if a stable speed distribution is achieved at low speeds, and once the velocity stabilizes, most model parameters can be determined. At each time step, the residual error is 10⁻³. A stability analysis of the time step was performed, and a time step of 0.025 s was determined.

The current study uses the Reynolds-Averaged Navier–Stokes (RANS) framework with the Realizable k-ε turbulence model, which provides a reliable balance between computational efficiency and engineering accuracy for predicting the statistical properties of wind loads (e.g., average pressure and key vortex-shedding frequency) on large-scale structures. It is recognized that although RANS models are robust for such applications, capturing the full range of turbulent fluctuations and complex vortex behavior requires more computationally intensive methods, such as Large Eddy Simulation (LES) or high-fidelity Direct Numerical Simulation (DNS)-type models.

Recent advancements in high-fidelity CFD have shown that high-order numerical methods combined with adaptive mesh refinement (AMR) can achieve better resolution of vortex-dominated flows at benchmark levels. For example, Zou et al. [20] developed a high-order discontinuous Galerkin (DG) scheme with AMR for vortex-induced vibration (VIV) simulations. Their work on a basic cylinder case demonstrated the ability of high-resolution methods to clearly capture vortex structures and closely match benchmark data. This highlights the complementary roles of different CFD approaches: high-fidelity methods like those in Zou et al. [20] establish a standard for resolving detailed flow physics in fundamental setups, while practical RANS-based simulations, as used here, offer an efficient and reliable tool for studying the main wind effects on complex, real-world historical structures like the Yingxian Pagoda, where parametric studies and long transient analyses are often necessary.

2.3. Grid Partitioning

Computational meshes are mainly classified as structured or unstructured. While unstructured meshes offer greater flexibility and geometric adaptability, enabling efficient mesh refinement, they require careful balancing of the element count. Insufficient mesh density reduces solution accuracy, whereas overly dense meshes increase computational time and can cause numerical vibrations [20]. This study used a hybrid polyhedral–hexahedral mesh for grid generation. Local mesh refinement was applied to improve the computational model’s accuracy, as shown in Figure 3a. This hybrid meshing approach effectively captures complex geometric features, contributing to more physically representative and reliable simulation results. Additionally, mesh quality (measured on a scale from 0 to 1) is vital for convergence and accuracy of results. The resulting mesh has a minimum quality of 0.43 and an average of 0.95, indicating high-quality discretization suitable for numerical simulation. To assess mesh independence, monitoring lines (Figure 3b) were placed downstream of the pagoda to extract average velocity profiles. Velocity variations for mesh sizes of 0.5, 1.0, 1.5, 2.0, and 2.5 million cells were analyzed. As shown in Figure 3c, when the mesh reaches 1.5 million cells, fluctuations in the mean wind velocity stay within 0.7%. Therefore, a mesh size of 1.5 million cells was chosen for all subsequent simulations to balance accuracy and computational efficiency. This grid-convergence method follows established CFD guidelines to ensure mesh-independent results, which are essential for reliable simulations, as emphasized in recent methodological reviews [21].

A specific focus was placed on refining the mesh near the wall boundaries (pagoda surface and ground) to ensure accurate capture of boundary layer effects within the RANS framework. The dimensionless wall distance, y+, for the first cell centroid next to the walls was kept within a range of 30 to 300, with an average around 85. This range was intentionally chosen to support the use of standard wall functions employed by the Realizable k-ε turbulence model. A y+ value in this logarithmic region means the viscous sublayer is not directly resolved but is instead modeled by the wall function, offering a computationally efficient and accurate representation of near-wall turbulence and shear stress in high-Reynolds-number flows typical of wind engineering applications.

2.4. Boundary Conditions and Model Parameters

In the CFD model, a velocity inlet condition was applied at the upstream boundary, while a pressure outlet condition is used downstream. The latter is employed when both the outlet pressure and flow velocity are unknown, assuming fully developed flow, in which the normal gradient of all flow variables is zero [20]. The surfaces of the wooden tower and the ground are defined as no-slip walls, setting both velocity and turbulence intensity to zero. The top and side boundaries of the computational domain are treated as free-slip symmetry planes, simulating unconfined flow conditions. According to the local government’s meteorological report, Ying County has a typical northern temperate continental climate with distinct seasonal changes. Southwesterly airflows mainly influence the region throughout the year. Seasonal monsoons occur mostly in winter and spring, with prevailing winds coming from the southwest. The average annual wind speed is 2.4 m/s, and the maximum wind speed can reach 20 m/s. Therefore, five simulation models were created for inlet velocities ranging from 10 m/s to 30 m/s, with a turbulence intensity of 5% and a turbulent viscosity ratio of 10%. Other computational model settings are shown in

Table 1. Computational model settings.

Computational Models	Parameter Settings
Solver	SIMPLE Second Order of Pressure First Order Upwind of Turbulent Kinetic Energy Transient
Turbulence Model	Realizable k-ε
	Standard Wall Function
Gravity	−9.81 m/s2
Inlet Boundary Types	velocity inlet
Inlet velocity magnitude	10 m/s 15 m/s 20 m/s 25 m/s 30 m/s
Wall Conditions	No-slip wall at the bottom free-slip wall at the sides and top
Convergence Error	0.0001

. Additionally, all numerical simulations were performed on the Sugon supercomputing platform (Sugon, Beijing, China).

2.5. Numerical Model Validation: Benchmark Testing of 2D Cylinder Flow

The geometric structure of the Yingxian Wooden Pagoda is of such complexity that its aerodynamic behavior under strong winds can be considered to fall within the realm of large-scale bluff-body flow and flow separation. In order to verify the accuracy and reliability of the CFD numerical methods adopted in this study, a benchmark validation on the classic flow past a two-dimensional (2D) circular cylinder (d = 0.6 m) was conducted prior to the pagoda simulation.

As demonstrated in Figure 4a, adequate distances are established for the boundaries of the computational domain to efficiently prevent interference from blockage, and a high-precision boundary layer mesh is generated in the near-wall region. The numerical configuration employed in this validation was meticulously consistent with that utilized in the Yingxian Wooden Pagoda simulations, encompassing the Realizable k-ε turbulence model, transient solver, SIMPLE algorithm, convergence criteria, and meshing strategy.

The present study investigates the aerodynamic forces in a cross-wind setting, focusing on the vortex-induced forces. These forces are extracted at varying wind speeds, ranging from 10 to 30 meters per second. The simulation results demonstrate that, following an initial transient excitation phase, the aerodynamic forces manifest as constant-amplitude sinusoidal periodic oscillations. This phenomenon is exemplified in Figure 4b. The statistical analysis of the stable segments reveals that the time-averaged means approach zero infinitely, thereby confirming that the flow field is completely convergent. Furthermore, by comparing the periods extracted from the CFD simulations with the theoretical periods calculated based on the classical Strouhal relationship (which will be detailed subsequently in Section 3.3; see Formula (3)), the error analysis indicates that under all wind speed conditions, the relative errors are kept within an extremely small range (the maximum error is only 3.6%, decreasing to 0.80% at high wind speeds).

In summary, contemporary CFD numerical simulation methods have been demonstrated to possess the capability to reliably capture fundamental physical phenomena, including flow separation and wake vortex shedding, with a high degree of precision. Consequently, the extension of the application to the three-dimensional complex shape wind field simulation of the Yingxian Wooden Pagoda is both reasonable and highly credible.

3. Results and Analysis

3.1. Flow Field Structure and Pressure Distribution near the Pagoda

Taking a model with a flow velocity of 20 m/s as an example, we analyze the unsteady flow characteristics around the wooden tower. Figure 5 shows the evolution of the flow field structure at T = 9.78 s, including the velocity cloud on the monitoring surface depicted in Figure 3b and the isosurface plot of 3D velocity magnitude = 10 m/s. The velocity around the Pagoda decreases due to the obstruction. As shown in Figure 5(a1)–(d1), the influence area in front of the Pagoda is a standard circular shape, while the influence area behind it forms a triangular region with the Pagoda at the apex. In the leeward region, flow separation occurs, leading to an alternating vortex pattern consistent with a Kármán vortex street. The periodic generation and shedding of vortices in the wake are clearly visible within a full cycle T. From the 3D isosurface plots in Figure 5(a2)–(d2), the spatial structure of the separated flow field is more distinctly observed: it extends from the Pagoda as the starting point to both sides at a certain angle. It gradually slopes downward from the top of the Pagoda to the ground. Moreover, at the rear end of the separated flow, independent shedding structures are visible near the ground, indicating the Kármán vortex street.

Similarly, the flow field structures at different wind speeds are shown in Figure 6. As the inflow wind speed increases, the maximum flow velocity in the field also increases. When the inlet velocity reaches 10 m/s, a separated flow forms behind the tower, but no clear shedding structure is observed. As the inlet velocity increases further, an independent shedding structure of the Kármán vortex street develops closer to the tower (as indicated by the red line in the figure). Notably, the angle of the triangular shape at the rear of the tower remains nearly unchanged with increasing flow velocity, increasing slightly from 18.88° to 20.39°.

Figure 7 illustrates the surface wind pressure distribution contours on the pagoda at the XY plane (Z = 10 m height) at various inlet velocities. It is shown that as the incoming flow velocity increases, the maximum wind pressure on the structure’s surface increases as well. The highest pressure is concentrated on the windward facade due to the tower volume directly obstructing the incoming flow. The region’s geometric symmetry and direct exposure to the flow make it particularly prone to flow stagnation effects, resulting in maximum wind pressure loads. The simulation results suggest that the points of maximum wind pressure are located on both sides of the front windward wall. According to the Bernoulli principle, the increase in wind pressure within this positive pressure zone can be approximately represented as:

$$\Delta p={\displaystyle \frac{1}{2}}\rho (v^2-v_{\mathrm{r}\mathrm{e}\mathrm{f}}^2)$$

(2)

In bluff-body flow problems, the reference velocity is usually defined as the freestream velocity in a region far enough from the object that the flow remains undisturbed. When the inlet wind speed rises from 10 m/s to 25 m/s, the theoretical dynamic pressure, which depends on the square of the velocity, increases by 525%. As shown in Figure 7, numerical simulation results reveal that the maximum pressure zone is concentrated on both sides of the windward face. This distribution pattern aligns with the flow mechanism around an octagonal cross-section bluff body, where flow separation occurs at the corners, forming localized high-pressure regions. Conversely, higher dynamic-pressure conversion efficiency occurs in flat-wall areas due to attached flow.

3.2. Vortex-Induced Vibration Analysis of the Yingxian Wooden Pagoda

When a steady flow surrounds a bluff body, two rows of parallel linear vortices with opposite rotational directions periodically form on each side of the object. This flow pattern develops into a Kármán vortex street, driven by nonlinear interactions. Under such flow conditions, the alternating vortex shedding causes fluctuating forces, leading to vortex-induced vibration (VIV) of the structure. Specifically, the shedding vortices create fluctuating surface pressures in both the in-line and cross-flow directions. These pressure fluctuations can induce significant vibrations, potentially compromising the integrity and stability of the structure. Therefore, analyzing the characteristics of these pressure fluctuations is crucial for protecting the structure.

In the case of the Yingxian Wooden Pagoda, the fluctuating pressures on its surface are mainly due to the asymmetric wind-pressure distribution induced by airflow around the structure. It is clear that the obstruction created by the pagoda leads to a specific pressure pattern on the surface, as the airflow moves around the structure. As shown in Figure 8, the pressure distribution on the tower’s surface varies significantly with wind speed: the windward side experiences positive pressure that increases with wind speed. Due to boundary-layer separation, a negative-pressure zone forms on the side. As wind speed rises, the upper part of the negative pressure zone expands considerably more than the lower part. The pressure behind the tower remains negative.

The time-history curves of the forces, as shown in Figure 9, clearly display periodicity in the wind (x), crosswind (y), and vertical (z) directions. First, there are no periodic fluctuations in the tower forces when wind speeds are below 10 m/s. However, when wind speeds exceed 15 m/s, the forces exhibit significant fluctuations. Second, the cross-wind (y) force exhibits a notably stable period, and its frequency aligns with the vortex-shedding frequency observed in the flow-field visualization (Figure 9), consistent with classical aerodynamic theory. Third, a key finding is that the pulsation period along the wind direction (x) is synchronized with the period along the side direction (period ratio ≈ 1). This differs from the frequency doubling expected for a streamlined cylinder. This unusual response, combined with the pagoda’s unique octagonal cross-section, strongly suggests significant differences in its flow separation and reattachment mechanisms. Additionally, a strong periodic force is observed in the vertical (z) direction, closely synchronized with the vortex-shedding process. Flow field analysis confirms that this force results from the aerodynamic effects caused by the multi-layer eaves under unsteady flow conditions. This systematic revelation and mechanistic analysis of the three-dimensional vortex-induced forces provide a crucial theoretical foundation for accurately assessing the wind-induced dynamic response and safety performance of the Yingxian Wooden Pagoda and similar ancient timber structures.

The quantitative models of these forces, particularly the dominant cross-wind component, are crucial not only for safety assessment but also for guiding the development of targeted mitigation strategies. For example, recent studies have examined non-destructive environmental measures that follow a quadratic function, such as strategically placed sheltering trees, to lessen wind loads on this pagoda [18]. The effectiveness of these methods largely depends on a detailed understanding of the baseline vortex-shedding patterns and force regimes established in this study.

The statistical results shown in

Table 2. The Maximum, minimum, amplitude, and period of vortex-induced forces of the pagoda in the x, y, and z directions at different wind speeds.

		Wind Speeds
		10 m/s	15 m/s	20 m/s	25 m/s	30 m/s
Drag force ($x$-dir)	Maximum/kN	62.366	142.729	254.965	405.673	588.240
	Minimum/kN	62.354	142.400	253.838	400.703	574.233
	Mean/kN	62.360	142.564	254.200	403.146	581.000
	Amplitude/kN	0.005	0.165	0.436	2.470	7.004
	Cycle time/s	18.92	12.74	9.77	7.65	6.15
Vortex-Induced force ($y$-dir)	Maximum/kN	0.000837	0.637	3.556	4.727	15.008
	Minimum/kN	−0.00562	−0.635	−2.797	−4.723	−16.149
	Mean/kN	0.00169	0.00163	0.442	−0.219	−0.129
	Amplitude/kN	0.00323	0.636	3.176	4.725	15.578
	Cycle time/s	18.40	12.78	9.79	7.68	6.23
Lift force ($z$-dir)	Maximum/kN	48.900	110.617	199.012	313.452	456.354
	Minimum/kN	48.763	109.329	196.699	303.435	431.213
	Mean/kN	48.831	109.978	198.001	308.907	443.892
	Amplitude/kN	0.068	0.762	1.157	5.009	12.570
	Cycle time/s	18.90	12.70	9.78	7.88	6.00

demonstrate that the amplitude and period of the vortex-induced forces on the pagoda in the x, y, and z directions change regularly with wind speed: The mean and amplitude of the drag and lift forces increase nonlinearly and rapidly. The mean of the vortex-induced force remains around zero, and the amplitude also rises quickly. The forces in all three directions share the same vibration period and decrease as inflow wind speed increases. To quantitatively describe this variation and explore the underlying nonlinear dynamic features, curve fitting was performed on the amplitude and period data from Table 2 against wind speed. The results are displayed in Figure 10, where Figure 10a shows the periodic time results, and Figure 10b–d show the force results in the y-, x-, and z-directions, respectively. The cycle time is inversely proportional to inlet velocity. The amplitudes in all three directions grow exponentially with increasing inlet velocity. The mean values of the drag and lift forces follow a quadratic function with respect to inlet velocity, with an R² value of 0.9999. Substituting a speed of 0 m/s into the fitting function in Figure 10 causes the vibration cycle times of all forces to tend to infinity, and their mean and amplitude values approach zero. This confirms that the fitting function accurately reflects the general flow characteristics.

To assess the potential resonance risk, the vortex-shedding frequencies from the current simulations were compared with the pagoda’s natural frequencies. As shown in the dynamic characteristic analysis of the Yingxian Wooden Pagoda [3], the first two fundamental natural frequencies are approximately 0.33 Hz (first bending mode) and 0.88 Hz (first torsional mode). In contrast, the dominant vortex-shedding frequencies are defined by f = 1/T, where T is the period listed in Table 2. The frequency range identified in this study spans from 0.06 Hz (at 10 m/s) to 0.16 Hz (at 30 m/s), across the simulated wind speeds. This comparison clearly demonstrates a separation between the main vortex-excitation frequencies and the structure’s key natural frequencies within the wind speed range of 10–30 m/s. Therefore, the likelihood of classical “lock-in” resonance, where the shedding frequency matches a natural frequency, is considered low under these conditions. However, it is important to note that the periodic vortex-induced forces observed in this study, even at non-resonant frequencies, are the primary mechanism driving the pagoda’s dynamic response. The ongoing action of these cyclic loads can lead to accumulated fatigue damage in the ancient timber joints and components, significantly affecting the long-term deformation and safety of the structure [22]. Consequently, quantifying these forces is crucial for conducting a fatigue-damage-oriented safety assessment.

3.3. Relationship Between Vibration Amplitude of the Wooden Pagoda and Wind Speed

In order to understand the potential hazards caused by vortex shedding to the pagoda, it is necessary to determine the frequency of the vortex street formed during the process. The vortex-shedding frequency is determined by the $\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{h}\mathrm{a}\mathrm{l}$ formula. That is:

$$S_{\mathrm{r}}={\displaystyle \frac{fd}{v}}$$

(3)

where $S_{\mathrm{r}}$ represents the $\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{h}\mathrm{a}\mathrm{l}$ number, $f=1/T$ and is the characteristic frequency of the periodic flow, $d$ is the characteristic length, and $v$ is the incoming flow velocity. The $\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{h}\mathrm{a}\mathrm{l}$ number is a dimensionless parameter that is related to the cross-sectional shape. The structure factor, which can be estimated as a constant, is typically around 0.14 [23]. However, this value is based on an ideal, infinitely long cylinder and therefore cannot be directly applied to the pagoda-shaped tower with a specific height and shape in this study. The characteristic length d is taken as 30.27 m for the Yingxian Wooden Pagoda, and the theoretical cycle times under different wind speeds are then calculated and compared with the numerical results, with a deviation of approximately −11%.

Furthermore, based on the fitting results shown in Figure 10a, the Strouhal number of Yingxian Wooden Pagoda is $0.16$. In order to continue using the classical Strouhal number 0.14 to describe the flow around a cylinder, a correction factor of 0.875 was introduced into the Strouhal Formula (3) to consider the specific shape characteristics of the Yingxian Wooden Pagoda. The modified formula is given as follows:

$$S_r=0.875{\displaystyle \frac{fd}{v}}$$

(4)

The corrected theoretical cycles are listed in

Table 3. Comparison of theoretical and simulated vortex-shedding cycles and relative errors at different wind speeds.

Wind Speeds	10 m/s	15 m/s	20 m/s	25 m/s	30 m/s
Theoretical cycle/s	21.621	14.414	10.811	8.649	6.443
Simulated cycle/s	18.740	12.740	9.780	7.740	6.127
Corrected Theoretical cycle/s	18.919	12.613	9.459	7.568	6.306
Relative Errors	−0.95%	1.00%	3.38%	2.28%	−2.84%

, and all relative errors are less than $\pm 4\%$.

The shape-specific correction coefficient of 0.875 was introduced for the Yingxian Wooden Pagoda’s distinctive octagonal cross-section. This approach is justified by recent findings that the Strouhal number varies systematically with the corner modification rate of a structural section [24]. Therefore, applying a dedicated shape correction to this non-standard octagonal profile is both reasonable and necessary.

4. Conclusions

This study uses CFD numerical simulation to analyze the wind pressure distribution, vortex-induced vibration characteristics, and methods for assessing the wind resistance of the Yingxian Wooden Pagoda. The main conclusions are as follows:

(1)

A detailed description of the verification process and simulation method for the CFD technique is provided for the wooden pagoda structure. A transient simulation approach utilizing the Realizable k-εε turbulence model and an unstructured mesh achieves high accuracy while significantly improving computational efficiency, serving as a reference for low-cost, high-efficiency “digital wind tunnel” evaluations.

(2)

The periodic shedding of the Kármán vortex street behind the pagoda and the variation in flow structures at different inlet wind speeds are examined. The simulations show a wind pressure distribution characterized by a high-pressure area on the windward side and a large low-pressure area on the leeward side, driven by boundary-layer separation. The vortex-induced force in the cross-wind (y-direction) is confirmed as a key driver of the structural dynamic response.

(3)

The simulations systematically reveal and quantify the periodic vortex-induced forces acting on the Yingxian Wooden Pagoda. These are fluctuating forces in all three directions when the velocity exceeds 10 m/s, with amplitudes increasing exponentially as the inlet velocity rises. The average values of drag and lift forces follow a quadratic relationship with inlet velocity.

(4)

The results clearly demonstrate that the traditional Strouhal formula can be adapted by using a correction factor of 0.875, resulting in a vortex-shedding frequency prediction model suitable for this type of ancient pagoda. This offers a quantitative tool for wind-vibration safety evaluations and wind-resistant design of similar wooden structures. Importantly, the correction factor (0.875) was specifically derived considering the octagonal cross-section and geometric proportions of the Yingxian Wooden Pagoda. Its primary application is for octagonal timber pagodas. This shape-specific approach aligns with recent studies indicating that the Strouhal number varies systematically with changes in the corner geometry of bluff bodies. Therefore, applying this factor to pagodas with different cross-sectional shapes (e.g., square, circular, or hexagonal) is not advised, as vortex-shedding dynamics may differ significantly.