Prediction of Pier Scour Depth under Extreme Typhoon Storm Tide

Abstract

The Western Pacific region is highly vulnerable to typhoon storm surge disasters, with localized erosion posing a particularly prominent issue for coastal marine structures. The prevalence of extreme typhoon storm surges poses a significant threat to the safety of engineering projects in these areas. In this study, a parameterized wind field model with precise calculation of wind speed was employed to establish a numerical model for typhoon storm tides. Based on the Western Pacific typhoon data from 1949 to 2023, hydraulic simulations were conducted for Hangzhou Bay, Xiangshan Port, and Yueqing Bay, revealing maximum flow velocities of 4.5 m/s, 1.95 m/s, and 2.09 m/s, respectively. These velocities exceeded the maximum possible tidal flow by 0.47–1.17 m/s. Additionally, using Sun’s velocity formula, the initiation flow velocities were calculated to be 1.85 m/s, 1.81 m/s, and 2.06 m/s for the aforementioned locations. Through localized erosion tests conducted around typical bridge piers and the subsequent application of similarity criteria, the maximum depth of localized erosion in the study area was determined to range from 2.16 m to 16.1 m, which corresponds to 1.1–2.3 times the scour caused by the maximum tidal flow scenario. A comparison of the erosion test results with calculations based on several formulas demonstrated that the scour prediction formula proposed by Sun exhibited the highest accuracy. This study supplements the understanding of the impact of typhoon storm surges on bridge pier erosion and provides a scientific basis for the design of bridge foundations.

1. Introduction

The storm surge caused by typhoons has a significant impact on the near-field and far-field currents of the bridge. The frequency and intensity of typhoons show an increasing trend with global warming [1]. Under the influence of typhoon storm currents, there may be an increasing trend in the local scour depth around bridge piers. Therefore, it is crucial to calculate the scouring of bridge piers under the action of extreme typhoon surges.

Based on numerical simulations and prototype measurements of the Shanghai Yangtze Bridge, Tao et al. [2] conducted an analysis of local scouring around bridge piers under typhoon 9711 and flooding. However, there is relatively little research on the impact of typhoons on bridge pier erosion, while there is more research on the impact of tsunamis on bridge pier erosion [3,4] or natural erosion and deposition changes caused by typhoons [5,6]. For example, Li et al. [7] used a fully coupled hydrodynamic and sediment transport CFD model to simulate the flow field and erosion around bridge piers caused by 14 tsunamis. The results showed that the tsunami period, T, and peak velocity, Um, determined the erosion rate. At the same time, the shear stress on the bed increased due to streamline bending. However, tsunamis are a series of long wave propagation caused by enormous energy generated by crustal movement. As they approach land, natural disasters—such as shallow water depth causing water surface rise and increased flow velocity—may have similar effects on the surface as typhoon surges; however, their mechanisms and causes are completely different and cannot be generalized.

The storm surge caused by typhoons is the main factor causing serious marine disasters in China, and numerical simulation of typhoon surges is extremely important for marine disaster prevention and reduction [8]. According to statistical data presented by Lu et al. [9], the East China Sea experiences an average of four typhoon events per year. In the past, there has been a focus on the flooding and breach of embankments caused by storm surges [10,11,12] but there has been little research on the erosion around building foundations caused by storm surges. Due to the vast sea area involved in typhoons, parameterized models are required for simulating typhoon surges to input into the typhoon field. Sun et al. [13] found that typhoons are affected by sea surface resistance during their movement, and existing wind field models do not consider the influence of sea surface resistance, resulting in a significant deviation between the wind speed field and reality. Wind speed has a significant effect on storm current velocity, which in turn affects local erosion on bridge piers; therefore, it is necessary to choose a parameterized wind field model with higher accuracy in wind speed field.

Estuarine, coastal, and harbor regions, predominantly composed of cohesive silty sands, are significantly affected by typhoons. Extensive research has been conducted on scour around bridge piers in non-cohesive sandy riverbeds [14,15], with recent studies increasingly focusing on scour processes in non-cohesive sandy substrates in coastal areas [16]. Scouring results from the interaction of water flow with the bed surface, and the initiation of sediment transport is often studied using the critical flow velocity, especially when addressing the scouring of actual structures.

Table 1. Unified formulas for the critical flow velocity of sediment.

Author	$\mathbf{Formula}$
Sha (1956) [17]	$u_c=0.74\left[\mathit{lg}\left(11{\displaystyle \frac{H}{K_s}}\right)\right]{\left({\displaystyle \frac{{\rho }_s-\rho }{\rho }}gd+0.19{\displaystyle \frac{gH\delta +{\epsilon }_k}{d}}\right)}^{\frac{1}{2}}$
Zhan (1961) [18]	$u_c={\left({\displaystyle \frac{H}{d}}\right)}^{0.14}{\left(17.6{\displaystyle \frac{{\rho }_s-\rho }{\rho }}d+6.05\times {10}^{-7}{\displaystyle \frac{10+H}{d^{0.72}}}\right)}^{\frac{1}{2}}$
Dou (2001) [19]	$u_c=0.74\left[\mathit{lg}\left(11{\displaystyle \frac{H}{K_s}}\right)\right]{\left({\displaystyle \frac{{\rho }_s-\rho }{\rho }}gd+0.19{\displaystyle \frac{gH\delta +{\epsilon }_k}{d}}\right)}^{\frac{1}{2}}$
Sun (2007) [20]	$u_c={\displaystyle \frac{8}{7}}{\displaystyle \frac{1}{{\sigma }_d^{1/8}}}{\left[{\displaystyle \frac{{\rho }_s-\rho }{\rho }}g\sqrt{d{d}_k}+0.06{\left({\displaystyle \frac{{\rho }^{\prime }}{{\rho }_s^{\prime }}}\right)}^3{\displaystyle \frac{g\left(H+h_0\right)d\delta }{d_k^2}}\right]}^{1/2}{\left({\displaystyle \frac{H}{K_s}}\right)}^{1/6}$

Formulas involve the variables as follows: ${\rho }_s$ is the particle density; ${\rho }^{\prime }$ is the wet density of sediment; ${\rho }_s^{\prime }$ is the dry density of sediment; d is the median grain size of sediment particles; $H$ is the water depth; δ is the thickness of the water film; e is the porosity; ${\epsilon }_k$ is the adhesion parameter; $K_s$ is the bed roughness; g is the acceleration due to gravity; ξ is a coefficient related to relative exposure; $d_{\mathrm{k}}$ is the average grain size of the $\mathrm{k}$ non-uniform sand class; δ is the thickness of the surface layer; ${\sigma }_d$ is the standard deviation of non-uniform sand; and $h_0$ is the additional head height.

presents a unified formula for the critical flow velocity applicable to both coarse- and fine-sediment particles. The differences among these formulas stem from variations in cohesive forces and are derived from perspectives of force or torque balance. Sha (1956) [17] attributed the cohesive force to the unidirectional transmission of hydrostatic pressure by thin water films. Zhang’s formula (1961) [18] considers the influence of porosity on adhesive forces. Dou (2001) [19] incorporated the additional downward pressure of water on particles and the dry bulk density of sediments. Sun (2007) [20] derived a formula for the initiation of non-uniform cohesive sands based on a combination of probabilistic and mechanical approaches.

There have been numerous studies on the prediction formula for bridge pier erosion, mostly based on empirical or semi-empirical methods [21,22,23,24,25,26,27,28,29]. The above formulas are mostly based on indoor experimental data, and the main factors include water depth, pier diameter, the Froude number, and relative flow velocity. However, under extreme hydrodynamic conditions such as typhoon storm that occur once in a century or more, the water flow velocity in natural sea areas is far greater than the starting velocity of sediment, driving sediment movement. At this time, natural erosion and local erosion coexist in the sea area where the marine structure is located, making the mechanism of pier erosion more complex, resulting in significant differences in the calculation results of each formula and increasing the uncertainty of pier safety. Therefore, it is necessary to collect local scour data of bridge piers under extreme typhoon surges to test the formulas and provide more recognized and reliable calculation methods for bridge piers under extreme conditions.

The western Pacific Ocean is severely affected by typhoon surge disasters, and coastal areas will inevitably face local scouring of bridge piers under the influence of typhoon surges, posing a threat to the safety of offshore structural engineering. Despite the increasing frequency of typhoons due to global warming, the academic focus has primarily been on tsunamis rather than typhoon-induced scour. This study focuses on the prediction of pier scour depth during extreme typhoon storm tide conditions. Consequently, the primary objectives of this study are as follows:

• A parameterized wind field model, incorporating more accurate wind speed calculations, is being considered for the development of a numerical model to study typhoon storm tide. By utilizing Western Pacific typhoon data spanning from 1949 to 2023, comprehensive simulations of extreme typhoon hydrodynamics will be conducted for Hangzhou Bay, Xiangshan Port, and Yueqing Bay. Through these simulations, the essential parameters of water depth and maximum flow velocity required for localized erosion experiments and formula derivation under extreme hydrodynamic conditions will be determined, facilitating a better understanding of the impact of such conditions on bridge pier erosion;
• Based on the soil samples from the bridge pier sections susceptible to erosion identified through erosion tests in various marine areas, the initiation flow velocities will be computed to establish a physically representative sand-based experimental bed for simulating extreme hydrodynamics associated with localized bridge pier erosion. This physical model aims to accurately measure the maximum depth of localized erosion occurring around the bridge piers;
• Four different formulas for calculating the depth of localized erosion around bridge piers will be employed. These formulas will be used to estimate the maximum depth of erosion under extreme hydrodynamic conditions for typical bridge piers in various marine areas. Subsequently, a comprehensive comparison will be conducted between the calculated values and the corresponding measured values. This analysis will provide essential technical parameters for the design of bridge foundations, offering valuable insights for bridge engineering.

2. Materials and Methods

2.1. Storm Tides Model

A mathematical model of the typhoon surge was constructed based on Delft3D, and the continuity equation is as follows:

Continuity equation

$${\displaystyle \frac{\partial \zeta }{\partial t}}+{\displaystyle \frac{1}{\sqrt{G_{\eta \eta }}\sqrt{G_{\xi \xi }}}}{\displaystyle \frac{\partial \left((d+\zeta )U\sqrt{G_{\eta \eta }}\right)}{\partial \xi }}+{\displaystyle \frac{1}{\sqrt{G_{\eta \eta }}\sqrt{G_{\xi \xi }}}}{\displaystyle \frac{\partial \left(\right(d+\zeta \left)V\sqrt{G_{\xi \xi }}\right)}{\partial \eta }}=0$$

(1)

Momentum equations in the $\xi$ and $\eta$ directions

$$\begin{array}{l}{\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{u}{\sqrt{G_{\xi \xi }}}}{\displaystyle \frac{\partial u}{\partial \xi }}+{\displaystyle \frac{v}{\sqrt{G_{\eta \eta }}}}{\displaystyle \frac{\partial u}{\partial \eta }}+{\displaystyle \frac{w}{d+\zeta }}{\displaystyle \frac{\partial u}{\partial \sigma }}={\displaystyle \frac{v^2}{\sqrt{G_{\eta \eta }}\sqrt{G_{\xi \xi }}}}{\displaystyle \frac{\partial \sqrt{G_{\eta \eta }}}{\partial \xi }}-{\displaystyle \frac{uv}{\sqrt{G_{\eta \eta }}\sqrt{G_{\xi \xi }}}}{\displaystyle \frac{\partial \sqrt{G_{\xi \xi }}}{\partial \eta }}\\ +fv-{\displaystyle \frac{1}{{\rho }_0\sqrt{G_{\xi \xi }}}}{P}_{\xi }+F_{\xi }+{\displaystyle \frac{1}{(d+\zeta {)}^2}}{\displaystyle \frac{\partial }{\partial \sigma }}\left({\nu }_V{\displaystyle \frac{\partial v}{\partial \sigma }}\right)+M_{\xi }\end{array}$$

(2)

$$\begin{array}{l}{\displaystyle \frac{\partial v}{\partial t}}+{\displaystyle \frac{u}{\sqrt{G_{\xi \xi }}}}{\displaystyle \frac{\partial v}{\partial \xi }}+{\displaystyle \frac{v}{\sqrt{G_{\eta \eta }}}}{\displaystyle \frac{\partial v}{\partial \eta }}+{\displaystyle \frac{w}{d+\xi }}{\displaystyle \frac{\partial v}{\partial \sigma }}={\displaystyle \frac{uv}{\sqrt{G_{\eta \eta }}\sqrt{G_{\xi \xi }}}}{\displaystyle \frac{\partial \sqrt{G_{\eta \eta }}}{\partial \xi }}-{\displaystyle \frac{u^2}{\sqrt{G_{\eta \eta }}\sqrt{G_{\xi \xi }}}}{\displaystyle \frac{\partial \sqrt{G_{\xi \xi }}}{\partial \eta }}\\ -fu-{\displaystyle \frac{1}{{\rho }_0\sqrt{G_{\eta \eta }}}}{P}_{\eta }+F_{\eta }+{\displaystyle \frac{1}{(d+\zeta {)}^2}}{\displaystyle \frac{\partial }{\partial \sigma }}\left({\nu }_V{\displaystyle \frac{\partial v}{\partial \sigma }}\right)+M_{\eta }\end{array}$$

(3)

where $\xi$ is the water level; $d$ is the reference plane depth in the model; $\sqrt{G_{\xi \xi }}$ and $\sqrt{G_{\eta \eta }}$ are the coordinate conversion coefficients; $u$, $v$, and $w$ are velocities in the $\xi$, $\eta$, and $\sigma$ directions; $U$ and $V$ are depth average velocities; ${\nu }_V$ is the vertical eddy viscosity coefficient; and $P_{\xi }$ and $P_{\eta }$ are the pressure gradients in the $\xi$ and $\eta$ directions. $F_{\xi }$ and $F_{\eta }$ are Reynolds stress terms in the $\xi$ and $\eta$ directions; and $M_{\xi }$ and $M_{\eta }$ are momentum source terms in the $\xi$ and $\eta$ directions; the typhoon wind speed is introduced from here.

This article adopts Sun‘s parameterized wind field model, which considers the sea surface resistance term as a parameterized wind field model. This model is not only theoretically more reasonable but also has been verified with the measured pressure and wind speed of typhoon Tracy (1974) and typhoon Kerry (1979), greatly improving the calculation accuracy of the wind speed profiles. Sun’s formula is as follows:

Pressure field mode:

$$P(r)=P_c+\left(P_{\mathrm{\infty }}-P_c\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{R}{r}}\right)$$

(4)

Gradient wind $V_g\left(r\right)$ and maximum wind $V_{\mathrm{m}\mathrm{a}\mathrm{x}}$ speed:

$$V_g(r)=\sqrt{{\displaystyle \frac{\left(P_{\infty }-P_c\right)R}{{\rho }_{a}r}}exp\left[-\left({\displaystyle \frac{R}{r}}\right)\right]+{\displaystyle \frac{{\left(f+3v/4{\delta }_a^2\right)}^2{r}^2}{4}}}-{\displaystyle \frac{\left(f+3v/4{\delta }_a^2\right)r}{2}}$$

(5)

$$V_{\mathrm{m}\mathrm{a}\mathrm{x}}=\sqrt{{\displaystyle \frac{\left(P_{\mathrm{\infty }}-P_c\right)}{{\rho }_{a}e}}+{\displaystyle \frac{{\left(f+3v/4{\delta }_a^2\right)}^2{R}^2}{4}}}-{\displaystyle \frac{\left(f+3v/4{\delta }_a^2\right)R}{2}}$$

(6)

where $P_c$ represents the central pressure within the typhoon, which can be obtained from the China Meteorological Administration (CMA) Best Track dataset; $P_{\infty }$ is the ambient pressure far from the typhoon, typically assumed to be 1013 hPa; R denotes the radius of maximum wind (RMW); and ${\delta }_a$ is the depth of the surface boundary layer.

The computational domain covers the entire East China Sea, coastal and estuarine waters at Zhejiang, as well as the Yangtze Estuary. The large-scale model meshes are set up as 385 × 600 × 6 with the resolution of 1500 m, while the small-scale model meshes have observation point S1 in Hangzhou Bay; the observation point S2 are set up with small-scale model meshes in Xiangshan Bay and the observation point S3 are configured with small-scale model meshes in Yueqing Bay. The models were initialized with a cold start, and the tidal elevation boundary at the open sea was obtained from the global ocean tidal prediction model TPXO. The eddy viscosity coefficient for the large-scale model was set to 80 m²/s, while for the small-scale model, it was set to 10 m²/s. Manning’s coefficient for the terrain transitioned slowly from 0.012 m^−1/3·s to 0.018 m^−1/3·s based on the topography.

2.2. Model Validation

Based on the existing storm surge measurement data collected from S1, S2, and S3, Typhoon 9417, which had a significant impact on the Yueqing Bay area in history, Typhoon 0205, which had a significant impact on Shangang, and Typhoon 2114, which had a significant impact on Hangzhou Bay, were selected for simulation verification.

Typhoon 9417 made landfall in Meitou, Rui’an City, Zhejiang Province at 2:30 am on August 21. As it was the 15th day of the lunar calendar, during the astronomical high tide period, both the Ou River and the Feiyun River experienced historical high tide levels. The Wenzhou high tide reached 7.21 m, which was 0.61 m higher than the highest tide level of 6.66 m in 1992 and close to the frequency of a 200-year return period. The highest tide level of the Feiyun River was 6.79 m, which was 0.38 m higher than the 6.41 m high tide level in 1992. The specific path is shown in Figure 1.

The path and process of Typhoon 0205 are shown in Figure 1. The minimum pressure of Typhoon 0205 was 950 hPa, the maximum wind speed was close to 45 m/s, and the distance between the typhoon center and the Chinese mainland was more than 200 km. Typhoon 2114 was located approximately 110 km northeast of Taiwan at 19:00 on 12 September, with its typhoon level changing from super typhoon to strong typhoon. Afterward, the typhoon moved northward and the wind gradually weakened. At 10:00 a.m. on 13 September, the offshore platform of the bridge entered a typhoon level 7 wind circle. At 17:00, the center of the typhoon was located on the east side of Zhoushan, closest to the monitoring site (approximately 200 km away). On 14 September at 2:00 p.m., Typhoon 2114 turned southeast after circling in the open sea, and a wind circle of magnitude 7 was introduced at the scene. As of 12:00 on the same day, the bridge was approximately 51 km away from the boundary of Typhoon 2114’s Category 7 wind circle.

The root mean square error (RMSE) represents the distribution of data points around the best-fit line. For a perfect match between calculated and measured values, the root mean square error is zero. The RMSE is obtained by the following formula.

$$RMSE=\sqrt{{\displaystyle \frac{\Sigma {\left(Q_{m_i}-Q_{c_i}\right)}^2}{M}}}$$

(7)

From the overall simulation results in the Figure 2, it can be seen that the RMSE tidal level validations S1, S2, and S3 are 0.13, 0.19, and 0.18 m, respectively, indicating that the established typhoon surge model in this study has a high degree of agreement with actual measurement data and can better reflect the typhoon process; therefore, this model can provide tidal conditions for the study of erosion of S1, S2, and S3 bridge piers under the influence of typhoon surges.

2.3. Experimental Setup

Taking into account factors such as sea depth, extreme flow velocity, model sand selection, bridge pier size, and seawall layout, a large-scale bridge pier scouring test was conducted in the wave tide pool of the Port Engineering Museum at Zhejiang University’s Zhoushan Campus. A 7 m long and 5 m wide wall was built along the north–south direction as the experimental area. We set up a flow-increasing and contraction wall to simulate the scouring of bridge piers under extreme conditions and widened the sedimentation tank to dissipate energy from water flow and settle sediment. The physical model mainly simulates the depth and approximate shape of local scouring pits on bridge piers, and the model can be designed according to normal conditions, that is, the plane scale and vertical scale are consistent. As shown in Figure 3, Figure 4 and Figure 5, the terrain and engineering structure around the bridge piers are scaled according to the drawings to simulate their water-blocking effect, local erosion range, and maximum erosion depth.

In the sand-laying section, a height of 40 cm was laid. After laying the selected model sand, the downstream tailgate was closed, and water was slowly injected into the water tank. It was left to stand for 48 h to allow the model sand to fully absorb the moisture. The experimental area after soaking is shown in Figure 6.

Before the experiment began, an aluminum bracket with a length of 5.5 m was erected between the two walls, and a sliding rail was fixed on the bracket. The ADV was placed on the bracket and slowly filled with water until the required water depth for the experiment was reached, and then the flow rate was gradually increased. At the same time, the tailgate was slowly opened, and the ADV could slide along the rail to measure the flow velocity at each position of the experiment. The underwater terrain measurement instrument using ultrasound was used to detect real-time elevation changes, and the depth of the scouring pit was observed and recorded every 10 to 20 min. When it was observed that the local erosion pit no longer deepened and expanded, the local erosion had reached an equilibrium state. Then, we closed the tailgate and shut down the water pump. After the flow rate dropped to 0, we slowly opened the tailgate to discharge the water from the water tank. We observed the local erosion pattern around the bridge pier after the water in the sink had drained. The observation of the erosion pit morphology used underwater terrain measurement with a range of 0.03 to 40 m, an accuracy of 0.1 mm, and an operating temperature of 0 to 60 degrees, which met the needs of the terrain measurement.

Due to the fact that the flow velocity during the typhoon surge is greater than the incipient flow velocity of the sediment, natural erosion will occur during erosion tests, and the bottom elevation will generally decrease. After natural erosion reaches equilibrium, due to the increase in water depth, the actual flow velocity decreases, while the incipient flow velocity increases. When the two approach, the bottom sediment will not move, and natural erosion under the storm surge reaches equilibrium. At this point, due to the action of the bridge piers, local erosion around the piers continues, resulting in local erosion pits. To reflect the joint effect of natural erosion and local erosion, the experiment adopts the following method: First, without placing the model, the calculated flow velocity during the once-in-a-century typhoon surge period was used for erosion. After the natural erosion was completed and the bottom bed stabilized, the water was drained and the model placed, and the original flow rate was continued for erosion. At that time, the local erosion results around the pier were obtained. After the local erosion around the model pier reached equilibrium, we drained the water and measured the local erosion around the pier to obtain the depth of the local erosion.

3. Results and Discussion

3.1. Determination of Extreme Typhoon Currents in the Sea Area

In most years, the East China Sea is affected by typhoons. According to typhoon paths, typhoons can be mainly divided into three categories: the first category is typhoons that make landfall in Zhejiang or Shanghai; the second category is typhoons that travel northward along the coast of Zhejiang; and the third category is typhoons that make landfall in Fujian and Guangdong.

This study collected historical typhoon path datasets provided by the World Meteorological Organization (WMO) and the China Meteorological Administration (CMA) and analyzed that the historical typhoon path data provided by the China Meteorological Administration (CMA) are the most comprehensive and accurate in the Western Pacific region. The histogram of the annual occurrence of typhoons from 1949 to 2023 is shown in Figure 7, based on intergenerational statistics. As shown in Figure 7, since 1949, a total of 1882 tropical cyclones have been recorded in the Western Pacific, including 426 typhoons, 340 severe typhoons, and 442 super typhoons.

The relationship between central air pressure and maximum wind speed radius was calculated over the years and it was found that the maximum wind speed radius in the Western Pacific Ocean is 180 km, with an average of 90 km [30]. We centered S1, S2, and S3, respectively, as shown in Figure 8, and drew circles with the typhoon path within a 180 km radius of the center of S1, S2, and S3 to analyze the impact of typhoons from 1949 to 2023. The specific paths of each typhoon are shown in the Figure 8. Within a radius of 180 km, an increasing trend is shown for tropical cyclones (TD, 10.8 m/s ≤ $V_{max}$ < 17.1 m/s), tropical storms (TS, 17.2 m/s ≤ $V_{max}$ < 24.4 m/s), severe tropical storms (STS, 24.5 m/s ≤ $V_{max}$ < 32.6 m/s), typhoons (TY, 32.7 m/s ≤ $V_{max}$ < 41.4 m/s), severe typhoons (STY, 41.5 m/s ≤ $V_{max}$ < 50.9 m/s), and super typhoons (SuperTY, $V_{max}$ ≥ 51.0 m/s). Within a 90 km and 180 km radius of the centers S1, S2, and S3, cyclones in regions S2 and S3 are equal, while the number of TS and TY in region S2 is more than in regions S2 and S3, as shown in Figure 9.

Based on actual measurement data from 1949 to 2023, the spatial distribution of the minimum pressure and maximum wind speed for each grid cell are illustrated in Figure 10. Figure 10 also shows that in the 180 km radius of centers S1, S2, and S3, the maximum air pressure and wind speed in each region correspond to Typhoons 5612, 9417, and 9711, respectively. Therefore, the aforementioned typhoons are simulated to obtain the significant impact on these regions in history. The maximum flow velocity parameters Vs and corresponding water depths for S1, S2, and S3 are listed in

Table 2. Flow parameters in positions S1, S2, and S3.

Location	Pier Type	${\mathbf{V}}_{\mathbf{s}}$	Water Depth
S1	D1–D3	4.5	15
S2	D4/D8	1.95	20.68
	D5/D9	1.95	20.57
	D6/D10	1.95	19
	D7/11	1.72	18.6
S3	D12–D13	2.09	4.96
	D14	2.09	6.56

. Among them, the pier type D1–D3 is a single pile, while the other D4–D14 adopt a group pile with a cap structure.

The maximum possible tidal current velocity around the engineering area is calculated using the following formula:

$${\overrightarrow{V}}_t=1.295{\overrightarrow{V}}_{M_2}+1.245{\overrightarrow{V}}_{S_2}+{\overrightarrow{V}}_{K_1}+{\overrightarrow{V}}_{O_1}+{\overrightarrow{V}}_{M_4}+{\overrightarrow{V}}_{{MS}_4}$$

(8)

where ${\overrightarrow{V}}_{M2}$, ${\overrightarrow{V}}_{S_2}$, ${\overrightarrow{V}}_{K1}$, ${\overrightarrow{V}}_{O1}$, ${\overrightarrow{V}}_{M_4}$, and ${\overrightarrow{V}}_{{MS}_4}$ represent the flow velocities corresponding to the tidal components of $M_2,S_2$, $K_1$, $O_1$, $M_4$, and ${MS}_4$, respectively. According to the harmonic analysis results, it can be concluded that S1, S2, and S3 may have maximum flow velocities of 2.72 m/s, 1.48 m/s, and 0.92 m/s. Similarly, we will present the difference between the maximum flow velocity $V_s$ of the storm current and the maximum possible flow velocity $V_t$ in

Table 3. The storm current and the maximum possible tidal velocity.

Location	Pier Type	${\mathbf{V}}_{\mathbf{t}}$	${\mathbf{V}}_{\mathbf{s}}-{\mathbf{V}}_{\mathbf{t}}$
S1	D1–D3	2.72	1.78
S2	D4–17	1.48	0.24/0.47
S3	D12–D13	0.92	1.17

.

3.2. Starting Similarity and Model Sand Selection

The numerical model employed in this study simulates tidal flow dynamics by adhering to the principles of gravity, resistance, and continuity while satisfying the surface tension and turbulence constraints. In simulating sediment transport, the model primarily follows the sediment incipient motion similarity criterion, with due consideration given to the settling similarity condition. The key dimensionless parameters, namely ${\lambda }_u$, ${\lambda }_{u_c}$, ${\lambda }_{\omega }$, ${\lambda }_t$, and ${\lambda }_{nb}$, respectively, represent the velocity scale, sediment incipient motion velocity scale, sediment settling velocity scale, flow timescale, and bed roughness scale.

Similarity conditions of fluid flow can be derived through the application of similarity theory, utilizing the two-dimensional non-steady flow continuity in Equations (1)–(3). Based on the aforementioned equations, the flow velocity similarity conditions for the Froude number can be derived:

$${\lambda }_u={\lambda }_v={\lambda }_l^{1/2}={\lambda }_h^{1/2}$$

(9)

where ${\lambda }_l$ and ${\lambda }_h$ are the geometric scale.

The S1 study focuses on large-diameter single-pile foundations with flow resistance widths of 2.5 m, 3.7 m, and 5.5 m for the bridge piers. The depth of the sea area is 10–20 m, with an average of 15 m. The maximum average flow velocity at the navigation channel during flood tide was measured to be 1.74 m/s, while the flow velocity near the offshore platform was significantly higher, reaching 2.72 m/s. The seabed sediment composition near the S1 station of the Hangzhou Bay Cross-Sea Bridge was primarily characterized by clayey silt and silty clay, with a median grain size ranging from 0.013 to 0.017 mm in

Table 4. Particle size near the S1 station of Hangzhou Bay Cross-Sea Bridge.

${\mathbf{d}}_{50}$ (μm)	${\mathit{\sigma }}_{\mathit{d}}$	Grain Size Composition (%)			${\mathit{u}}_{\mathit{c}}$ (m/s)
		250–75 μm	75–2 μm	<2 μm
17.8	1.85	6.1	85.9	8	1.86
13.14	1.87	4.5	86.1	9.4	1.6
15.46	1.81	4.5	87.2	8.3	1.71

. Based on the formula proposed by Sun et al. (2007) [20], the average critical erosion velocity was calculated to be 1.72 m/s.

According to the measured hydrological conditions in the region, a 1:50 geometric scale was used for the design of the water tank experiment, with model piers of 5 cm, 7.5 cm, and 11 cm, respectively.

Within a range of 20 m below the riverbed at the S2 bridge pier, silt and loam are mainly composed of clay (particle size less than 0.004 mm) and silt (0.004 mm to 0.063 mm). The clay content accounts for 50% to 60%, with a median particle size of 0.007–0.009 mm. The intergranular adhesion force dominates. Based on the smaller particle size of fine-grained mud and sand, it is more difficult to start. From the perspective of engineering safety, the median particle size of the prototype bottom sand is determined to be 0.009 mm.

The composition of the seabed (washable part) near S3 is relatively consistent. The washable riverbed in the engineering area is mainly composed of silty clay and clayey silt. The d50 at the bridge site is selected as the average value of 0.0059 mm. The intergranular viscous force dominates. According to Sun’s (2007) formula, the sediment threshold flow velocity for each region is calculated and listed in

Table 5. Model test incipient flow rate similarity.

Region	Geometric Scale	Flow Rate Scale	Prototype Value (m/s)	Design Value (m/s)	Model Sand Particle Size (mm)
S1	${\lambda }_l$ = 50	${\lambda }_{v_c}={\lambda }_u=7$	1.85	0.26	0.16
S2	${\lambda }_l$ = 100	${\lambda }_{v_c}={\lambda }_u=$10	1.81	0.18	0.12
S3	${\lambda }_l$ = 40	${\lambda }_{v_c}={\lambda }_u=6.32$	2.06	0.32	0.05

.

3.3. Comparison between Experimental Values and Formula Predictions

Traditional normative formulas, 65-2 equations, Han’s formula [31], and Sun’s formula [32] are used to calculate the scour depths of the bridge in positions S1, S2, and S3, respectively, to evaluate the hedging depth prediction effectiveness of each, seen in

Table 6. Calculation formula for maximum scouring depth around bridge piers.

Author	Formula	Remarks
65-2 equations	${\mathrm{d}}_{\mathrm{s}\mathrm{e}}={\mathrm{k}}_{\mathsf{\epsilon }}{\mathrm{k}}_{\mathsf{\eta }2}{\mathrm{D}}^{0.6}{\mathrm{H}}^{0.15}({\displaystyle \frac{\mathrm{u}-{\mathrm{u}}_{\mathrm{C}}^{\prime }}{{\mathrm{u}}_{\mathrm{c}}}}{)}^{{\mathrm{n}}_2}$ ${\mathrm{n}}_2={\left({\mathrm{u}}_{\mathrm{c}}/\mathrm{u}\right)}^{0.23+0.19\mathrm{l}\mathrm{g}\mathrm{d}}$ ${\mathrm{k}}_{\mathsf{\eta }2}=\left({\displaystyle \frac{0.0023}{{\mathrm{d}}^{2.2}}}+0.375{\mathrm{d}}^{0.24}\right)$ ${\mathrm{u}}_{\mathrm{c}}=0.28{\left(\mathrm{d}+0.7\right)}^{\frac{1}{2}}$ ${\mathrm{u}}_{\mathrm{C}}^{\prime }=0.12{\left(\mathrm{d}+0.5\right)}^{0.55}$	${\mathrm{d}}_{\mathrm{s}\mathrm{e}}$ is the scouring depth of the bridge pier; ${\mathrm{k}}_{\mathsf{\epsilon }}$ is the coefficient of influence of riverbed particles; ${\mathrm{k}}_{\mathsf{\eta }2}$ is the influence coefficient of riverbed particles; $\mathrm{H}$ is the maximum water depth; ${\mathrm{n}}_2$ is the index; ${\mathrm{u}}_{\mathrm{C}}^{\prime }$ is initiation scour velocity; ${\mathrm{I}}_{\mathrm{L}}$ is the liquid index; $\mathsf{\phi }$ is the angle of repose of sediment underwater; $\mathrm{P}$ is the porosity of sediment; ${\displaystyle \frac{\mathrm{f}\left({\mathrm{B}}_0/\mathrm{D}\right)}{\mathsf{\pi }}}$ is the constant that can be calibrated based on experimental data and the actual scour depth of bridge piers.
Code formula (hydrological survey and design of highway engineering in China)	$\begin{array}{c}\begin{array}{ccc}{d}_{se}=0.55{\mathrm{k}}_{\mathsf{\epsilon }}{D}^{0.6}{{\mathrm{H}}^{0.1}I}_{L}u& & H/D<2.5\end{array}\end{array}$ $\begin{array}{c}\begin{array}{ccc}{d}_{se}=0.83{\mathrm{k}}_{\mathsf{\epsilon }}{D}^{0.6}{I_L}^{1.25}u& & H/D\ge 2.5\end{array}\end{array}$
Sun’s formula	${\displaystyle \frac{{\mathrm{d}}_{\mathrm{s}\mathrm{e}}}{\mathrm{H}}}={{\mathrm{k}}_{\mathsf{\epsilon }}\left[{\displaystyle \frac{\mathrm{f}\left({\mathrm{B}}_0/\mathrm{D}\right)}{\mathsf{\pi }}}\right]}^{\frac{1}{3}}{\left({\displaystyle \frac{\mathsf{\rho }}{{\mathsf{\rho }}_{\mathrm{s}}-\mathsf{\rho }}}{\displaystyle \frac{\mathrm{t}\mathrm{g}\mathsf{\phi }}{1-\mathrm{P}}}\right)}^{\frac{1}{3}}{\mathrm{F}\mathrm{r}}^{\frac{2}{3}}{\left({\displaystyle \frac{\mathrm{D}}{\mathrm{H}}}\right)}^{\frac{1}{3}}$
Han’s formula	${\mathrm{d}}_{\mathrm{s}\mathrm{e}}=8.48{\mathrm{k}}_{\mathsf{\epsilon }}{\mathrm{D}}^{0.326}{\mathrm{U}}^{0.628}{\mathrm{H}}^{0.193}{\mathrm{d}}_{50}^{0.167}$

.

Based on the simulation of typical typhoon storm surges, the maximum storm current velocities and corresponding water depths for S1, S2, and S3 were obtained. A physical model of the local scouring of bridge piers was constructed in a wave tide moving bed water tank. Based on typical bridge pier layout schemes, local scouring tests were conducted on bridge piers under the action of typhoon currents, and the depth of local scouring was measured as shown in the

Table 7. Local scour depth test values.

Location	Pier Type	${\mathbf{d}}_{\mathbf{s}\mathbf{t}}$ (m)	${\mathbf{d}}_{\mathbf{s}}$ (m)	${\mathbf{d}}_{\mathbf{s}}$$/{\mathbf{d}}_{\mathbf{s}\mathbf{t}}$
S1	D1	3.70	8.51	2.3
	D2	4.50	10.66	2.4
	D3	6.10	11.60	1.9
S2	D4	10.20	15.9	1.6
	D5	9.00	14.8	1.6
	D6	6.90	9.1	1.3
	D7	5.50	6.2	1.1
	D8	11.13	16.1	1.4
	D9	8.97	14.8	1.6
	D10	6.87	8.5	1.2
	D11	5.50	6.4	1.2
S3	D12	2.30	4.88	2.1
	D13	1.20	2.16	1.8
	D14	2.80	5.08	1.8

. Among them, ${\mathbf{d}}_{\mathbf{s}\mathbf{t}}$ is the maximum possible tidal current test value, and ${\mathbf{d}}_{\mathbf{s}}$ is the test value under storm surge. The maximum local scouring depth obtained through S1 experimental observation and conversion based on similarity criteria is 8.51–11.6 m, which is 1.9–2.3 times the maximum scouring depth under the current situation. The maximum local scouring depth around S2 is 6.2 m–16.1 m, and the corresponding maximum scouring depths around D4–D7 are 10.2 m, 9 m, 6.9 m, and 5.5 m, respectively; D8–D11 is 16.1 m, 14.8 m, 8.5 m, and 6.4 m, respectively, which is 1.1–1.6 times the maximum tidal erosion. Due to the shallow water depth, the maximum scouring depth of D12–D14 is 2.16–5.08 m, which is 1.8–2.1 times that of the maximum tidal erosion.

The predicted values of various formulas and the local erosion depth results measured by experiments are listed in the

Table 8. Calculation of local scour depth for each formula and bias.

Location	Pier Type	${\mathbf{d}}_{\mathbf{s}}$ (m)	Code Formula (m)	Bias (%)	65-2 (m)	Bias (%)	Han’s Formula (m)	Bias (%)	Sun’s Formula (m)	Bias (%)
S1	D1	8.51	8.32	−2.21	5.32	−37.49	8.14	−4.35	8.87	4.19
	D2	10.66	10.53	−1.20	6.73	−36.84	9.25	−13.20	10.10	−5.18
	D3	11.60	13.36	15.14	8.54	−26.40	10.53	−9.26	11.53	−0.59
S2	D4	15.9	12.87	−20.06	15.17	−5.76	9.99	−37.95	12.02	−25.33
	D5	14.8	11.62	−21.47	13.75	−7.07	9.55	−35.47	11.60	−21.63
	D6	9.1	7.68	−9.64	9.10	7.01	7.64	−10.08	9.26	8.90
	D7	6.2	4.49	−29.92	2.87	−55.10	5.25	−18.04	5.88	−8.18
	D8	16.1	13.81	−12.58	16.28	3.06	10.38	−34.29	12.50	−20.86
	D9	14.8	11.62	−21.47	13.75	−7.07	9.55	−35.47	11.60	−21.63
	D10	8.5	9.80	7.65	11.60	27.49	8.72	−4.14	10.60	16.44
	D11	6.4	4.49	−27.66	2.87	−53.65	5.25	−15.40	5.88	−5.22
S3	D12	4.88	4.75	−2.64	3.04	−37.81	4.29	−12.04	4.86	−0.48
	D13	2.16	1.97	−8.75	1.26	−41.71	2.66	23.21	2.98	37.91
	D14	5.08	4.78	−5.83	3.07	−39.64	4.35	−14.37	4.97	−2.18

, where Bias = $y_c\times 100/y_m - 1$. It can be seen that the calculation results of each formula are different, with Bias having a minimum of −0.48% and a maximum of −55.10%.

The comparison between the predicted values from various formulas and the measured values is shown in Figure 11, Figure 12, Figure 13 and Figure 14, where the 45° solid line represents the ideal line. Let r be the ratio of the calculated value to the measured value, and the two dashed lines represent r = 1.25 and r = 0.8. The standardized formula, Han’s formula, and Sun’s formula all account for 73% of the data points with values falling into $\mathrm{r}\in \left(\mathrm{0.8,1.25}\right)$, indicating that these have high accuracy. The scouring value of the bridge pier calculated using the 65-2 formula is significantly lower than the experimental value. The standard calculation results show that except for the calculated erosion value of S2 below 5 m, which is smaller than the experimental value, the performance is better, especially for the high accuracy of S1 prediction results, The calculated values of Han’s formula and Sun’s formula are distributed on both sides of the measured results. Except for the calculated erosion value above 10 m, which is significantly lower than the experimental value, the other study areas perform well.

The average relative error value (MNE) is used to further compare and evaluate the predicted values of various formulas with experimental results, where the MNE calculation formula is as follows:

$$MNE={\displaystyle \frac{1}{n}}{\sum }_{k=1}^n\left|y_c/y_m-1\right|$$

(10)

The MNE calculation results for each region are listed in the

Table 9. Average relative error values (MNE values) of various formulas.

Formula	S1	S2	S3
Normative value	6.18	18.8	11.25
65-2	33.57	20.7	39.86
Han’s formula	8.94	23.9	12.98
Sun’s formula	3.32	16	10.26

, and it can be seen that the MNE value of S1 calculated by Sun’s is the lowest with the highest accuracy, followed by the standardized formula. The MNE value calculated using Han’s formula within S2 is 23.9%; the MNE value calculated using the 65-2 formula is 20.7%; the MNE value calculated using the standard value is 18.8%; the MNE value calculated using Sun’s formula is 16%; and the MNE values calculated within S2 are all relatively large, ranging from 16% to 23.9%. The MNE value of the S3 internal standard calculation result is 11.25%; the MNE value of the 65-2 formula calculation result is 39.86%; the MNE value of Han’s formula calculation result is 12.98%; the MNE value of Sun’s formula calculation result is 10.26%; and the MNE value of the S3 internal calculation result is 10.26% to 11.25%.

4. Conclusions

The main conclusions are as follows:

(1)

Taking into account the water depth, extreme flow velocity, model sand selection, bridge pier size, and experimental venues in the East China Sea, a physical model design for a clear water moving bed was carried out. The physical model scales for S1, S2, and S3 are 50, 100, and 40, respectively, and using Sun’s incipient flow velocity formula, the incipient flow velocities of the regional prototype sand were calculated to be 1.85 m/s, 1.81 m/s, and 2.06 m/s, respectively. Natural sand with a median particle size of 0.16 mm and wood chips with a density of 1.2 t/m³ were determined as the S1, S2, and S3 model sand, meeting the starting similarity condition. The results can provide a technical basis for engineering design;

(2)

According to the typhoon path, pressure, and wind speed from the data of the China Meteorological Administration, a total of 1882 tropical cyclones affecting the Western Pacific from 1949 to 2023 were counted, including 426 Typhoons, 340 Severe Typhoons, and 442 Super Typhoons. There were 8, 7, and 7 super typhoons within a radius of 90 km centered on S1, S2, and S3, respectively. A total of 25 hypothetical typhoon paths were designed, and the center pressure of a 100-year typhoon in Hangzhou Bay was determined to be 914.1 mba. We simulated Typhoons 5612, 9417, 9711, 2106, 2212, and 2114, which all had a serious impact on the study area. The maximum storm current velocities for S1, S2, and S3 were 4.5 m/s, 1.95 m/s, and 2.09 m/s, respectively. The maximum possible currents were 2.72 m/s, 1.78 m/s, and 1.48 m/s, respectively, and the flow velocity increased by 0.47–1.17 m/s. This result provides a certain reference for risk assessment and disaster prevention and reduction of marine engineering in local waters;

(3)

Based on the mathematical model, the maximum flow velocity parameters and water depth results for each sea area are given, combined with the terrain and sediment characteristics of the sea area where typical bridge piers are located, and considering the most unfavorable situation, local erosion tests are carried out around typical bridge piers in each sea area. According to the similarity criterion, the maximum local erosion depth around S1 is calculated to be 8.51–11.6 m, which is 1.9–2.3 times the maximum current situation erosion; the maximum local erosion depth around S2 is 6.2 m–16.1 m, which is 1.1–1.6 times that of the maximum tidal erosion; the maximum local erosion depth around S3 is 2.16–5.08 m, which is 1.8–2.1 times the maximum tidal erosion. This result provides a scientific basis for the safety design of bridge piers. According to the observation results of the local scouring test in the sand tank model and comparisons with multiple formulas, the four formulas are close to the experimental results and all have high accuracy. Among them, Sun‘s theoretical scouring formula has the smallest MSE value, more concentrated data, and the highest accuracy. Norms, Han‘s, and 65-2 formulas have the second-highest accuracy. Sun‘s formula predicts that the maximum scouring depths for S1, S2, and S3 are 11.53 m, 12.5 m, and 4.86 m.