Numerical Simulation Method for the Rain-Wind Induced Vibration of the Three-Dimensional Flexible Stay Cable

Zichen, Pan; Jian, Guan; Jian, Wang; Canchen, Wang; Xiaoyu, Guo

doi:10.1115/1.4067384

Abstract

Under specific wind and rain conditions, the cable may exhibit low frequency and large amplitude phonomenon, which is called the rain-wind induced vibration (RWIV). Because the RWIV phenomenon encompasses interactions among gas, liquid, and solid phases, make it difficult to reproduce through numerical simulation method. In the previous research, based on lubrication theory, researchers have established theory models of two-dimensional (2D) coupled and rigid segment of three-dimensional (3D) cables for RWIV. Based on the previous research, the new numerical simulation method for RWIV of 3D flexible cable is proposed in this paper, which can reflect the interaction among the gas, liquid, and solid phases. The aerodynamics forces are calculated using comsol. The evolution of water film is calculated using matlab. In the previous studies, the cable deformation was not considered. Hence, cable vibration is derived from the solution of a single-degree-of-freedom vibration equation. In this study, the cable is treated as a deformable body. The vibration response of the cable under external force is solved using ansys. The bidirectional coupling effect between the cable and the water film is considered. The morphology of water film on the cable surface is changing with time, which causes the variation of the lift over time. Therefore, it is necessary to discretize the time so that the lift acted on the cable at each time-step can be obtained. The Restart technique in ansys is used for this purpose. Additionally, the transmission of information can be achieved by using matlab is used to control comsol and ansys. The rain-wind induced vibration phenomenon of 3D flexible cable is reproduced by using the new numerical simulation method, analyzing the variation of water film, cable lift, cable vibration under the influence of cable deformation. The mechanism of rain-wind-induced vibration is revealed again by analysing the 3D model. In addition, the advantages and necessity of the 3D cable vibration model are demonstrated by comparing it with the 2D model simulating cable vibration

1 Introduction

Cable-stayed bridges, a modern type of composite bridge, have experienced rapid development over the past two centuries due to their exceptional resilience, wind resistance. However, the rigid and the damping of cable are small. Under the combined action of wind and rain, the cable may exhibit low-frequency, high-amplitude vibration, which is called rain-wind induced vibration (RWIV) by researchers [1]. The RWIV phenomenon was first observed by Japanese researchers Hikami and Shiraishi in 1998 [2]. Subsequently, RWIV has been observed on many bridges around the world, including the Erasmus Bridge in the Netherlands [3], the Fred Hartmann Bridge in Texas, USA [4], the Dongting Lake Bridge in Hunan [5], and the Yangpu Bridge in Shanghai, China [6]. RWIV may cause the cable fatigue, reducing the durability and service life of the cables and potentially endangering the safety of the entire cable-stayed bridge.

Researchers have used four main research methods to conduct extensive research for the RWIV phenomenon: field observation, wind tunnel testing, theoretical analysis, and numerical simulation.

Field observation serves as a direct method for gathering authentic and reliable data. For instance, Geurts et al. observed RWIV at Erasmus Bridge in 1996 and mitigated the vibrations by attaching a PE cable between the cable and the bridge deck [3]. Main and Jones monitored the Fred Hartman Bridge in Texas, USA, for over a year. They found that moderate rainfall combined with wind speeds ranging from 4 m/s to 14.5 m/s could induce RWIV [7]. Zuo et al. further monitored RWIV on the Fred Hartman Bridge. They found that RWIV occurred in cable modes 2–6 at the wind speed between 5 and 10 m/s [4]. RWIV can be more systematically analyzed by conducting the wind tunnel test. Matsumoto et al. investigated the effects of cable inclination angle, surface material, rainfall, and wind deflection angle [8–13]. Yamaguchi et al. measured the lift, drag forces on cables in the wind tunnel test, concluding that the circumferential oscillation of the upper rivulet is crucial for the significant amplitude cable vibrations [14]. Similarly, Jing et al. studied the vibration response of the cable under different wind speeds and identified the interaction between the evolution of the upper rivulet and cable vibrations as the primary cause of RWIV [15–17]. Theoretical analysis is also pivotal in understanding RWIV. Based on Yamaguchi's 2-DOF model, Peil and Nahrath established a 3-DOF model, considering the geometric and physical nonlinearities of the cable, and pointed to the periodic evolution of the upper rivulet as the primary cause of RWIV [18]. Lv et al. established a two-dimensional (2D) single-degree of-freedom cable segment model to explore the impact of various significant variables on cable vibration [19]. Additionally, Gu et al. established a theoretical RWIV model for three-dimensional (3D) continuous elastic cable by deriving the 3D motion differential equation of the cable [20,21].

The development of computer technology has significantly enhanced the role of numerical simulation for studying RWIV. Lubrication theory, in particular, has proven essential in simulating the formation and oscillations of rivulets on cable surfaces. Lemaitre et al. established a 2D theoretical model based on lubrication theory. They used the wind pressure coefficient and wind friction coefficient, C_p (θ) and C_f (θ), to replace the effects of airflow and studied the evolution of water film on the cable surface [22–24]. Considered the variation of wind pressure coefficient and wind friction coefficient with time, Taylor and Robertson refined Lemaitre's model and obtained the evolution of water film on the horizontal static cable surface [25–27]. However, these researches only studied the evolution of water film on the cable surface. The cable vibration is not studied. Extending this research, Bi et al. integrated the models from Xu et al. [28] and Taylor to formulate 2D coupled equations of water film evolution and cable vibration. They studied the relationship between the water film, cable lift, and cable vibration [29–31].

Wang et al. further developed a 2-DOF model of RWIV based on lubrication theory, investigating the characteristics of water film evolution, cable vibration, and air aerodynamic forces [32]. Wang et al. established four different models to simulate the RWIV phenomenon [33]. Among these, the 2-DOF gas–liquid–solid model was identified as the most accurate [34]. Additionally, Bi et al. used the finite difference method to solve the evolution equation of water film and comsol to calculate the wind pressure coefficient C_p (θ) and wind friction coefficient C_f (θ) [34]. However, these simulations were primarily two-dimensional.

In a significant advancement, based on lubrication theory, Bi et al. formulated a 3D evolution equation for the water film. The 3D water film evolution can be obtained by solving this equation [35]. Nevertheless, the impact of the evolution of water film on the aerodynamic force was not considered. Addressing this problem, Wang et al. established a theoretical model of 3D water film evolution to study the impact of water film evolution on the aerodynamic force and cable vibration [36]. They later considered the influence of the cable vibration on the evolution of water film, establishing a 2-way coupled 3D model [37]. However, due to the short length of the cable in the test, the deformation of cable was not considered.

In the new wind tunnel test of Gao et al. [38], they utilized a significantly larger wind tunnel and a longer cable. The observed cable vibration had a great difference with the previous wind tunnel test. They considered the cable deformation, capturing the vibrational response and dominant frequencies of a 3D flexible cable under different wind speeds. Their findings emphasized the limitations of using 2D cable vibration model to simulate the 3D vibrations in the previous simulations. The spatial shape of cable cannot be obtained and the cable deformation is also not considered in the research of using 2D cable vibration model.

Consequently, a theoretical model for RWIV of a 3D flexible cable is established in this paper. The 3D flexible cable model established in ansys is used to simulate cable vibration accurately. The integration of matlab and comsol with ansys has been effectively achieved using the ansys Restart technology, facilitating information transfer between these softwares and establishing a novel numerical simulation method.

The theoretical model, solution methodology, and fundamental parameters of the numerical simulation are detailed in Sec. 2. Section 3 presents and analyzes the results obtained from the numerical simulation, including water film evolution, cable lift, and cable variation. Finally, the findings from the numerical simulation study are summarized in Sec. 4.

2 Model

RWIV as a complex phenomenon involves the liquid, gas, and solid phases. In this paper, these three phases are represented by the water film, aerodynamic forces, and cable vibrations. Therefore, matlab is used to calculate the evolution of water film on the cable surface. comsol is used to calculate the wind pressure coefficient and wind friction coefficient. ansys is used to simulate cable vibration response.

2.1 Evolution Equation of Water Film.

The cable with a radius of R and a horizontal inclination angle of α(0 deg ≤α ≤ 90 deg) is affected by gravity and horizontal wind with the speed of U, as shown in Fig. 1(a). The wind yaw angle is β(0 deg≤β ≤ 90 deg). The gravity is divided into two components: the gravitational component along the axial direction of cable g_Z, and the component in the A–A cross section g_N. The cylindrical coordinate system (e_r, e_θ, z) is adopted, as shown in Fig. 1(b)

g_{N} = g cos α

(1a)

g_{Z} = g sin α

(1b)

Fig. 1

View large Download slide

Model of stay cable (a) Spatial position of the stay cable and (b) diagram of the cylindrical coordinate system

The airflow is divided into two components U_N and U_Z, too.

U_{N} = \frac{U cos β}{cos δ}

(2a)

U_{Z} = \sqrt{U^{2} - U_{N}^{2}}

(2b)

The angle between the wind speed and the gravity component in the cross section is ψ = δ + π/2.

δ = arctan (sin α \times tan β)

(3)

Based on the lubrication theory, a continuous water film is assumed on the cable surface. The forces to which the water film is subject are listed in Fig. 2. The coordinates of any point in the water film are (r, θ, z) R ≤ r ≤ R + h, where h is the thickness of the water film. The velocity of the water film u is expressed as the component form u = u_re_r+u_θe_θ+u_ze_z. Consequently, the Navier–Stokes equations can be written as follows:

\begin{matrix} ρ (\partial_{t} u_{r} + u_{r} \partial_{t} u_{r} + \frac{u_{θ}}{r} \partial_{θ} u_{r} + u_{z} \partial_{z} u_{r} - \frac{u_{θ}^{2}}{r}) \\ = ρ g e_{r} - \partial_{r} p + μ {\partial_{r} [\frac{1}{r} \partial_{r} (r u_{r})] + \frac{\partial_{θ}^{2} u_{r}}{r^{2}} + \partial_{Z}^{2} u_{r} - \frac{2 \partial_{θ} u_{θ}}{r^{2}}} \end{matrix}

(4a)

\begin{matrix} ρ (\partial_{t} u_{θ} + u_{r} \partial_{t} u_{θ} + \frac{u_{θ}}{r} \partial_{θ} u_{θ} + u_{z} \partial_{z} u_{θ} - \frac{\partial_{r} u_{θ}}{r}) \\ = ρ g e_{θ} - \partial_{θ} p + μ {\partial_{r} [\frac{1}{r} \partial_{r} (r u_{θ})] + \frac{\partial_{θ}^{2} u_{θ}}{r^{2}} + \partial_{Z}^{2} u_{θ} - \frac{2 \partial_{θ} u_{r}}{r^{2}}} \end{matrix}

(4b)

\begin{matrix} ρ (\partial_{t} u_{Z} + u_{r} \partial_{t} u_{Z} + \frac{u_{θ}}{r} \partial_{θ} u_{Z} + u_{z} \partial_{z} u_{Z}) \\ = ρ g e_{Z} - \partial_{Z} p + μ {\partial_{r} [\frac{1}{r} \partial_{r} (r u_{Z})] + \frac{\partial_{θ}^{2} u_{Z}}{r^{2}} + \partial_{Z}^{2} u_{θ}} \end{matrix}

(4c)

\frac{1}{r} \partial_{r} (r u_{r}) + \frac{1}{r} \partial_{θ} u_{θ} + \partial_{Z} u_{Z} = 0

(4d)

Fig. 2

View large Download slide

Forces of the water film around the cable

The boundary conditions of the equation include:

Displacement boundary condition:
The water is relatively static to the stay cable at the bottom of the water film
${u_{r} |}_{r = R} = {u_{θ} |}_{r = R} = {u_{Z} |}_{r = R} = 0$
(5)
Stress boundary condition:
On the water-air interface $F (r, θ, z, t) = R + h (θ, z, t) - r = 0$ ⁠, the jump in the normal

shear stress is balanced by the water film surface tension

(σ_{g} - σ) n = k γ n

(6a)

σ = - p I + [μ \nabla u + {(\nabla u)}^{T}]

(6b)

σ_{g} = - p_{g} (θ) + τ_{g}

(6c)

T is the time, μ is the dynamic viscosity coefficient of water, γ is the surface tension, σ and σ_g are the stress tensors of water film and air, I is the identity tensor, n is the normal vector of the water-air interface, K is the curvature of the water film surface; P_g is the air pressure on the water film surface; τ_g is the viscous viscosity tensor of air.

Free boundary condition:

On the water–air interface, the water film satisfies the equation as follows:

\frac{D F}{D t} = \partial_{t} h - u_{r} + \frac{\partial_{θ} h}{r} \cdot u_{θ} + \partial_{Z} h u_{Z} = 0

(7)

The dimensionless 3D film motion equations can be obtained, based on Eqs. – and the three boundary conditions for performing dimensionless processing. The meaning of various parameters is listed in Table 1.

\partial_{\hat{t}} \hat{h} = - \frac{1}{{\hat{R}}_{C} + \hat{h}} {\begin{matrix} \partial_{θ} {\begin{matrix} - [\begin{matrix} ({\hat{g}}_{N} cos (θ - δ) - \hat{\ddot{y}} cos θ) ({\hat{R}}_{C} + \hat{h}) + \partial_{θ} \hat{k} + ({\hat{g}}_{N} sin (θ - δ) - \hat{\ddot{y}} sin θ) \partial_{θ} \hat{h} \\ + \partial_{θ} (\frac{ρ_{g} U_{N}^{2}}{2 \sqrt{γ ρ g}} C_{p} + \frac{ρ_{g} U_{N}^{2}}{2 \sqrt{γ ρ g}} C_{f θ} + \frac{ρ_{g} U_{N}^{2}}{2 \sqrt{γ ρ g}} C_{f z}) - \frac{5 ρ_{g} U_{N}^{2}}{4 \sqrt{γ ρ g}} C_{f θ} \end{matrix}] \frac{{\hat{h}}^{3}}{{\hat{R}}_{C}} \\ + \frac{3 ρ_{g} U_{N}^{2}}{4 \sqrt{γ ρ g}} C_{f θ} {\hat{h}}^{2} \end{matrix}} \\ + \partial_{\hat{Z}} {\begin{matrix} {3 \hat{g}}_{Z} ({\hat{R}}_{C} + \hat{h}) {\hat{h}}^{3} + \frac{ρ_{g} U_{N}^{2}}{2 \sqrt{γ ρ g}} C_{f z} (\frac{3}{2} {\hat{R}}_{C} {\hat{h}}^{3} + 2 {\hat{h}}^{3}) \\ - [({\hat{g}}_{N} sin (θ - δ) - \hat{\ddot{y}} sin θ) \partial_{Z} \hat{h} + \partial_{Z} \hat{k} + \partial_{Z} (\begin{matrix} \frac{ρ_{g} U_{N}^{2}}{2 \sqrt{γ ρ g}} C_{p} + \frac{ρ_{g} U_{N}^{2}}{2 \sqrt{γ ρ g}} C_{f θ} \\ + \frac{ρ_{g} U_{N}^{2}}{2 \sqrt{γ ρ g}} C_{f z} \end{matrix})] {\hat{R}}_{C} {\hat{h}}^{3} \end{matrix}} \end{matrix}}

(8)

Table 1

The meanings of various parameters

γ

Surface tension coefficient of water in air

ρ

Density of water

P_g

Air pressure on the surface of the water film

g

Acceleration of gravity

U_N

Vertical component of wind speed

ρ_g

Air density

t_{g θ}

Circumferential vectors along the cable at the junction of water film and air

C_{f Z} = \frac{t_{Z}}{\frac{1}{2} ρ_{g} U_{N}^{2}}

Wind friction coefficient along the cable axis

t_{g Z}

Axial tangential vectors along the cable at the junction of water film and air

C_{f θ} = \frac{t_{θ}}{\frac{1}{2} ρ_{g} U_{N}^{2}}

Wind friction coefficient along the circumferential of the cable

K

Curvature of the surface of the water film

C_{p} = \frac{P_{g}}{\frac{1}{2} ρ_{g} U_{N}^{2}}

Wind pressure coefficient

μ

Water's dynamic viscosity coefficient

\ddot{y}

Cable acceleration

g_N

Gravitational acceleration component in the plane

g_Z

Axial acceleration component of gravity

R

Cable radius

h

Water film thickness

2.2 Equation of the Cable Lift.

Figure 3 shows the forces at any section (A–A) of the cable. The

F_{r} (θ, z)

and

F_{θ} (θ, z)

is the normal vector and the tangential force at the water film bottom (r = R).

{σ_{0} = {- p I + μ [\nabla μ + {(\nabla u)}^{T}]} |}_{r = R}

(9)

F_{r} (θ, z) = e_{r} \cdot σ_{0} \cdot e_{r} = {- p + 2 μ \partial_{r} u_{r} |}_{r = R}

(10a)

F_{θ} (θ, z) = e_{θ} \cdot σ_{0} \cdot e_{r} = {μ [r \partial_{r} (\frac{u_{θ}}{r}) + \frac{\partial_{θ} u_{r}}{r}] |}_{r = R}

(10b)

Fig. 3

View large Download slide

The force of stay cable

The cable lift Fy can be found by jointly substituting equations and into the following equations:

F_{y} = \int_{0}^{L} \int_{0}^{2 π} [F_{r} (θ, z) sin θ + F_{θ} (θ, z) cos θ] \cdot R_{C} d θ d z

(11)

2.3 The Computational Fluid Dynamics Model.

Computational Fluid Dynamics (CFD) is a powerful numerical simulation technology for studying complex wind-induced vibration problems. The wind friction coefficient C_f and the wind pressure coefficient C_p are significantly influenced by the morphology of the water film on the cable. Despite minimal changes in the morphology of water film, its impact on C_f and C_p is substantial. Therefore, the influence of the morphology of the water film on the C_f and the C_p is calculated by using comsol.

Lemaitre [23] assumed that a continuous water film exists on the cable surface during RWIV occurrences. Initially, the thickness of water film is uniform on the cable surface. Given the short time interval ΔT, the external morphology of the water film remains unchanged within each time-step. The calculation of C_p and C_f involves the following steps:

First, calculate the distance from the center of cable to the outer surface of the water film.
Then, construct a quasi-cylinder with its outer surface formed by the water film.
Finally, obtain C_p and C_f through steady-state computational analysis.

The computational domain for the flow field is depicted in Fig. 4, which is shaped as a rectangle with a length of 27.5 D and a width of 20 D. The left side is the air inlet, and the right is the outlet. The center of the cable is located at coordinate origin, 10 D from the inlet, 17.5 D from the outlet, and 10 D from the top and bottom boundaries. The air density is set at 1.225 kg/m³. The dynamic viscosity is set at 1.51 × 10⁻⁵Pa·s.

Fig. 4

View large Download slide

Flow domain configuration

Figure 5 illustrates the mesh division of the flow field, with a denser mesh near the quasi-cylinder surface. The smallest mesh layer is 0.0002 m. The y⁺≈2 meets the computational accuracy requirements [39]. Here, y⁺ represents the nondimensional wall distance, calculated using $y^{+} = y u_{r} / v$ ⁠, where y is the fluid velocity, u_r is the height of the first layer grid, and v is the kinematic viscosity. Given that the y and the v are knew, the height of the first layer grid u_r should be calculated by determining the size of the y⁺.

Fig. 5

View large Download slide

Mesh in the domain

To ensure computational efficiency, the mesh size is increased in areas farther from the cable. The Spalart–Allmaras turbulence model [40] is employed for the calculation.

The boundary conditions are as follows:

The velocity boundary conditions (U=U_N, V=the vibration velocity of the cable) are set at the left inlet and the top and bottom boundaries;
The pressure outlet boundary condition (P=0) is set at the outlet;
The lubrication boundary conditions are adopted on the quasi-cylinder surface: U=0, V=0;

2.4 Three-Dimensional Flexible Stay Cable Model in ansys.

In this section, the 3D flexible cable model is established in ansys. The cable vibration response can be simulated by conducting force analysis for the model. The simulation process is detailed as follows:

Based on Gao et al.'s wind tunnel test [38], the radius, length, linear density of 3D flexible cable model are set to 0.04918 m, 8 m, and 1.03 kg/m, respectively. The actual cable model is not entirely made of steel in Gao et al.'s wind tunnel test [38], but is composed of multiple materials, as shown in Fig. 6. Therefore, after our repeated debugging, we set the moment of inertia (IZZ) and the moment of inertia (IYY) of the 3D flexible cable model to 3.258 × 10⁻⁸. The basic parameters of cable model are listed in Table 2.

Fig. 6

View large Download slide

Cross section of the flexible cable model

Table 2

Basic parameters of cable model

Radius of cable (R)

0.04918 m

Length of cable

8 m

Linear density of cable

1.03 kg/m

Area

π*R² m²

Density of the cable (ρ)

6.6665 kg/m³

Moment of inertia (IZZ)

3.258 × 10⁻⁸ m⁴

Moment of inertia (IYY)

3.258 × 10⁻⁸ m⁴

The Z axis is the axial direction of the cable. The Y axis is the direction of cable vibration. The model comprises 43 nodes, including nodes 1 to 41, as well as nodes 101 and 141. The positions of node 1 and 101 are (0,0,0). The positions of node 41 and 141 are (0,0,8). The distance between other nodes is 0.2. The model comprises 42 elements. Elements 1 to 40 can be obtained by connecting the two adjacent nodes. Element 41 is obtained by connecting the node 1 and 101. Element 42 is obtained by connecting the node 41 and 141. The properties of nodes and elements are listed in Tables 3 and 4.

Table 3

The properties of nodes

Type	number	Real constant	Material properties
Beam 4	1	AREA, IZZ, IYY	Elastic modulus (ex) 2e11
Default settings		(Area, moment of inertia, moment of inertia)	Damping ratio (damp) 0.005
			Density (dens) ρ
			Principal Poisson's ratio (prxy) 0.3

Type	number	Real constant	Material properties
Beam 4	1	AREA, IZZ, IYY	Elastic modulus (ex) 2e11
Default settings		(Area, moment of inertia, moment of inertia)	Damping ratio (damp) 0.005
			Density (dens) ρ
			Principal Poisson's ratio (prxy) 0.3

Table 4

The properties of elements

Type	number	Real constant
Combin14	2	Spring constant K (1005)
Solution type K1 (Linear Solution)
DOF select for 1D behavior K2 (Torsional ROTX)
DOF select for 2D + 3D behavior K3 (3-D torsional)

The displacement constraints along the x, y, and z axes are set at nodes 1 and 41. The rotational restrictions around the y and z axes are at nodes 1 and 41. The cable can only vibrate at the direction of rotation around the x-axis.

Now, the 3D flexible cable model has been established. The loads are imported as follows. The source of load is the cable lift, which is generated by the morphological variation of water film. The number of elements is 40 in this cable model. So, the calculated cable lifts need to be imported on the corresponding nodes. The direction of lift is along the y-axis. According to Eq. (11), the lift can be calculated. In Eq. (11), we integrate 2 π along the θ direction and integrate the length of an element, which is 0.02 m, along the z-axis direction. The $F_{r} (θ, z)$ and $F_{θ} (θ, z)$ are obtained according to the evolution of water film at each node. Because the thickness of water film is changing with time, the lift acted on the cable is different and needs to be recalculated at each time-step. Therefore, the Restart function of ansys is used, which can keep the motion state of the cable and update the lift acted on the cable so that the force analysis of the cable model can be continuously proceeded. The complete transient analysis method is employed to force analysis for the cable model. The acceleration of gravity is set along the y-axis direction as the initial condition of the complete transient analysis. According to the spatial position of the cable, as shown in Fig. 1(a), the y-axis direction is perpendicular to the axial position of the cable but not to the ground. So, the acceleration of gravity on the y-axis direction is gcosθ m/s².

The first and second natural frequencies are obtained by conducting the modal analysis of 3D flexible cable model, as shown in Table 5, which is consistent with the wind tunnel test of Gao et al. [38].

Table 5

Natural frequency of the cable

Stage	1	2
Frequency (Hz)	2.348	4.425

2.5 Numerical Solution Flow and Basic Parameters.

The evolution equations for the water film and the cable lift equations are detailed in Secs. 2.1 and 2.2. The influence of morphological variation of water film on thewind friction coefficient C_f and wind pressure coefficient C_p is calculated using comsol in Sec. 2.3. The cable vibration response is simulated by the force analysis of 3D flexible cable model using ansys in Sec. 2.4. The transmission of information between the three phases is given in Fig. 7.

Fig. 7

View large Download slide

The relationship between the three phases

Based on lubrication theory, it is assumed that a continuous water film exists on the cable surface. Therefore, the wind cannot be directly acted on the cable. The specific interaction process at each moment is as follows:

Wind impacts the water film, altering its shape. The effect of wind on the water film is reflected by the wind pressure coefficient C_p and the wind friction coefficient C_f.
Changes in the water film induce variations in cable lift.
Changes in cable lift cause the cable vibration.
The vibrations of the cable, in turn, influence the evolution of the water film.
The evolution of water film will cause the morphological variation of water film, affecting the wind pressure coefficient C_p and the wind friction coefficient C_f.

2.5.1 The Flow of Numerical Solutions.

The computational process is shown in Fig. 8. Initially, the cable is static, with zero displacement, velocity, and acceleration. The initial thickness of water film, h₀, is set at 0.1 mm, uniform across the cable. The calculative steps are as follows:

Fig. 8

View large Download slide

Solution process of calculation

The initial morphology of water film on the cable surface and other boundary conditions are used to calculate wind pressure coefficient C_p and wind friction coefficient C_f on every section by turn in comsol.
Following the calculation of C_p and C_f, the new thickness of water film at every point can be obtained by solving the Eq. (9).
Cable lift for each section is calculated by solving Eq. (11).
The calculated cable lift is then input into the 3D flexible cable model established in ansys software to determine the cable vibration response. The specific methods are given in Sec. 2.4.

So far, the numerical solution flow in a single time-step has been given. The results of cable vibration include the displacement, speed, and acceleration of cable. The speed of the cable vibration is used as a boundary condition in the comsol model. The acceleration of the cable is input into the water film motion equation as the necessary parameter.

2.5.2 Basic Parameter.

The RWIV phenomenon was successfully reproduced through the wind tunnel tests of Gao et al. [38]. They observed the maximum amplitude of cable vibration at a wind speed of 11.26 m/s, with an inclination angle α of 23.39 deg and a yaw angle β of 45 deg. The length of cable is 8 m. The relevant settings for the calculations are consistent with the wind tunnel test. Additional parameters used in the numerical simulation are listed in Table 6.

Table 6

Parameters for numerical simulation

Radius of cable R

0.04918 m

Natural frequency of the cable f₀

2.347 Hz

Acceleration of gravity g

9.8 kg/s²

Water density

ρ

1.0 × 10³ kg/m³

Kinetic viscosity coefficient of water n_u

1 × 10⁻⁶ m²/s

Surface tension coefficient of water in air

γ

0.072 N/m

Air density

ρ

_g

1.225 kg/m³

linear density of the cable

ρ

_s

1.03 kg/m

According to the results of Gao et al.'s wind tunnel test [38], the initial thickness of water film h₀ is assumed to be 0.1 mm in this paper. The total water volume on the cable surface is considered conservative.

Initially, the total water volume on the cable surface can be calculated by multiplying the initial thickness of the water film by the cable surface area. During the calculational process, the thickness of the water film at every point is constantly changing. To ensure the continuation of calculation, the maximum and minimum thicknesses of the water film are set at 1 mm and 0.01 mm, respectively. When the thickness of the water film at a certain point exceeds this maximum, it is adjusted down to 1 mm, with the reduced water assumed to drip along the cable. Conversely, when the thickness of the water film at a certain point falls below the minimum, it is adjusted down to 0.01 mm, with the additional water considered to be supplied by rainfall. Following these adjustments, the water volume at each point is summed up to compute the adjusted total water volume. If this adjusted total water volume is less than the initial total water volume, the missing water will be supplied to the upper surface of the cable. If the adjusted total water volume is larger than the initial total water volume, the excess water will be averagely distributed to each point of the cable and reduced accordingly. This process ensures the equilibrium and stability of the total water volume on the cable.

To determine an applicable time-step for simulations, the evolution of the water film at a wind speed of 11.26 m/s was calculated using various time steps, as depicted in Fig. 9. It can be found that the differences in water film morphology at the five kinds of time-step were minimal. Considering calculative accuracy and time cost, the time-step for numerical simulation is set at the 1 × 10⁻³ s.

Fig. 9

View large Download slide

The water film morphology calculated at different time steps

3 Numerical Simulation Result

3.1 The Evolution of Water Film.

The evolution of water film on the cable surface is studied in this section, including the formation of rivulet and time history and frequency spectrum analysis of the evolution of water film. Additionally, the circumferential and axial evolution of rivulets, are studied. The wind speed is maintained at 11.26 m/s. To provide a clear depiction of the evolution of water film, the 2-6 m segment of the cable is specifically analyzed.

3.1.1 The Formation of the Rivulet.

Figure 10 illustrates the thickness of the water film on the 3D flexible cable surface at t = 0.1 s and t = 0.2 s. The vertical axis represents the circumferential direction of the cable. The horizontal axis represents the axial direction of cable. The depth of color indicates the thickness of water film. Initially, a thin upper rivulet forms between the circumferential positions θ = 82.2 deg∼93.5 deg at 0.1 s. By 0.2 s, the upper rivulet becomes clearer, measuring approximately 0.15 mm in thickness and 9.7 mm in width.

Fig. 10

View large Download slide

The thickness of rivulet on the cable surface at 0.1 s and 0.2 s: (a) t = 0.1 s and (b)t = 0.2 s

The evolution of the upper rivulet within a cycle is presented in Fig. 11. Initially, the upper rivulet is relatively thick. As it reaches a certain thickness, the influence of aerodynamic forces is surpassed by gravity, causing the rivulet to flow downward. The reduction of water volume diminishes the action of gravity, making aerodynamic forces to predominate once more, leading to the reformation of the upper rivulet. The thickness of the upper rivulet at 4.2 s is similar with at 4.5 s, which indicates the periodicity of revolution of upper rivulet. The period of revolution of upper rivulet is about 0.3 s.

Fig. 11

View large Download slide

Rivulet thickness in a period: (a) t = 4.2 s, (b) t = 4.3 s, (c) t = 4.4 s, and (d) t = 4.5 s

3.1.2 Evolution of Water Film at Specific Positions.

Figure 12 displays the time histories and spectral analysis of evolution of water film thickness at five different positions along the cable. The z-coordinate values of five positions are respectively 2 m, 3 m, 4 m, 5 m, and 6 m. The circumferential angle is 78.75 deg. The five positions are located at the oscillational region of upper rivulets. Analysis of the time-history curves of water film thickness evolution reveals that the peak thickness on the cable surface occurs every 0.3 s, indicating a periodic evolution. Notably, the water film thickness is the largest at the midspan position. Spectral analysis shows that the dominant frequency of the evolution of water film is 2.8 Hz in five positions, closely approximating the natural frequency of the cable (2.347 Hz). It indicates that the evolution of water film on the cable surface is greatly affected by the natural frequency of cable during RWIV occurrences. The amplitude of revolution of water film thickness is also the largest at the midspan position. Moreover, the spectral analysis reveals a pronounced “frequency doubling” phenomenon in the variation of water film thickness. This phenomenon, previously identified in other studies [15,41,42], remains poorly understood and requires further investigation.

Fig. 12

View large Download slide

Time history and spectral analysis results of water film thickness at Z = 2 m–6 m, θ = 78.75 deg (a) Z = 2 m, θ = 78.75 deg, (b) Z = 3 m, θ = 78.75 deg, (c) Z = 4 m, θ = 78.75 deg, (d) Z = 5 m, θ = 78.75 deg, and (e) Z = 6 m, θ = 78.75 deg

3.1.3 Circumferential Evolution Analysis of the Rivulet.

Figure 13 illustrates the circumferential evolution of the water film on the cable surface at five different positions within 10 s. The z-coordinate values of five positions are respectively 2 m, 3 m, 4 m, 5 m, and 6 m. Several similarities are evident across the five positions. The formation timing of the rivulets is consistent; the lower rivulets form earlier than the upper rivulets at five positions. The formation locations of both lower and upper rivulets are similar at five positions. However, there is difference as well. As referenced in Fig. 2, the circumferential direction from 90 deg to 270 deg represents the leeward side, while circumferential direction from 270 deg to 0 deg and 0 deg to 90 deg represent the windward side. When the action of aerodynamic force is larger than gravity, the upper rivulet can accumulate at about 75 deg–90 deg on the upper surface of the windward side. When the action of aerodynamic force is exceeded by gravity, the water of upper rivulet begins to drop from the windward side. The amount of dropping water is different at five positions. The amount of dropping water is the largest at the midspan position.

Fig. 13

View large Download slide

(a) Z = 2 m at the Z = 2 m–6 m, circumferential change of water film on the cable surface, (b) Z = 3 m, (c) Z = 4 m, (d) Z = 5 m, and (e) Z = 6 m

3.1.4 Axial Evolution Analysis of the Rivulet.

Figure 14 presents the time-history curve of water film thickness variation on the cable surface at eight positions including two adjacent axial positions (Z = 4 m and Z = 4.2 m) and four adjacent circumferential positions (θ = 70.31 deg, θ = 73.19 deg, θ = 75.94 deg, and θ = 78.75 deg). The horizontal axis represents the time. The vertical axis represents the water film thickness. The black curve illustrates the variation in water film thickness at Z = 4 m, and the gray curve depicts the variation at Z = 4.2 m.

Fig. 14

View large Download slide

The time-history curve of water film thickness at eight positions: (a) θ = 70.31 deg, (b)θ = 73.19 deg, (c) θ = 75.94 deg, and (d) θ = 78.75 deg

To illustrate the variation in water film thickness more clearly, the curves between 6 and 7 s have been extracted, as shown in Fig. 15. It is observed that water film thickness does not increase or decrease monotonically. The main reason is that the water film is always in an unstable state due to the combined action of gravity, aerodynamic force, and surface tension. Additionally, although the time-history curves at adjacent axial and identical circumferential positions exhibit the similarity, the time of appearance of the maximum water film thickness has a slight difference. This variability indicates that the water film flows axially along the cable surface. Differences in the timing of peak thickness also occur between the same axial positions at adjacent circumferential locations. These observations suggest that water film on the 3D cable surface can flow along the axial direction and circumferential direction.

Fig. 15

View large Download slide

(a) θ = 70.31 deg between 6 and 7 s Variation in time-history curves of the water film thickness at eight positions between 6 and 7 s, (b) θ = 73.19 deg between 6 and 7 s, (c)θ = 75.94 deg between 6 and 7 s, (d) θ = 78.75 deg between 6 and 7 s

3.2 Variation of the Cable Lift.

The cable lift can be calculated using the Eq. (11). Figure 16 presents the time-history curves and the spectral analyses of cable lift variation at five different positions along the cable. The z-coordinate values of five positions are respectively 2 m, 3 m, 4 m, 5 m, and 6 m. The circumferential angle is 78.75 deg.

Fig. 16

View large Download slide

The results of the spectral analysis and time history of the cable lift at five axial positions (a) Z = 2 m, (b) Z = 3 m, (c) Z = 4 m, (d) Z = 5 m, and (e) Z = 6 m

Observations of the time-history variation of cable lift reveal similarities with the evolution of the water film at five axial positions. Firstly, the range of cable lift variation is greatest at the midspan position. Furthermore, the peak value of cable lift occurs approximately every 0.3 s, demonstrating the periodic nature of the cable lift. Spectral analysis confirms that the dominant frequency of cable lift variation at five positions is 2.8 Hz, consistent with the dominant frequency observed in the evolution of the water film. These findings indicate a significant correlation between the variations in cable lift and the evolution of the water film.

Figure 17 combines the time history curve of cable lift at the midspan position with an image depicting the circumferential evolution of water film. The horizontal axis represents time, the right vertical axis indicates cable lift, and the left vertical axis shows the circumferential position of the water film evolution, with film thickness represented by color depth. The time history of the cable lift is represented in red.

Fig. 17

View large Download slide

The phase difference between the evolution of water film and lift

When the water film is the thickest, the cable lift is the largest. When the water film thickness begins to thin, the cable lift also decreases. Almost no phase difference be found between the variation of cable lift and the evolution of water film, illustrating that the periodic circumferential evolution of the water film directly influences the variation in cable lift.

3.3 Response of the Cable Vibration.

The spatial shape of cable vibration within a cycle can be visualized using ansys. Additionally, the time-history curve and the spectral analyses of cable vibration is shown in Fig. 18.

Fig. 18

View large Download slide

The results of the spectral analysis and time history of the cable vibration at five positions: (a) Z = 2 m, (b) Z = 3 m, (c)Z = 4 m, (d) Z = 5 m, and (e) Z = 6 m

The displacement is largest at the midspan position, aligning with findings from Gao et al.'s wind tunnel test [38]. The cable vibration is also periodic. By observing the spectrum analysis results, the dominant frequency of cable vibration is also 2.8 Hz at five positions, which is close to the natural frequency of the cable. Besides, the dominant frequency of cable vibration is also consistent with that of water film evolution and cable lift variation. This suggests a phenomenon of frequency resonance between cable vibration, cable lift, and water film, which is consistent with the previous studies (Gao et al. [38], Jing et al. [15], Li et al. [16,43], Bi et al. [30,34,35], and Wang et al. [36,37]). This observation emphasizes a critical interrelationship between cable vibration, cable lift, and water film.

To analyze the relationship between cable vibration and cable lift, the time-history curves of both at the midspan position are presented in Fig. 19. The gray curve represents the cable vibration at Z = 4m, and the black curve represents the cable lift variation at Z = 4m. When the cable lift is the maximum, the cable vibrates to the lowest point. When the cable lift is the minimum, the cable vibrates to the highest point. There is a half-cycle phase difference between the cable lift and the cable vibration. Section 3.2 demonstrates almost no phase difference between the cable lift and the water film evolution. Consequently, it may be inferred that a similar half-period phase difference exists between the cable vibration and the water film evolution. In Fig. 20, the time history of the cable vibration is represented in red. Specifically, when the water film thickness is the least, the displacement of cable vibration is also the smallest. When the water film thickness is the largest, the displacement of the cable vibration is also the largest. This deduction is substantiated by combining the time-history curve of the cable vibration with the image of circumferential evolution of water film, as illustrated in Fig. 20. Ultimately, this analysis leads to a crucial conclusion: the cofrequency resonance between the cable vibration and the water film evolution plays a significant role in RWIV.

Fig. 19

View large Download slide

Phase difference between cable vibration and cable lift variation on the cable surface

Fig. 20

View large Download slide

Phase difference between the water film evolution and the cable vibration

3.4 Discussion.

Based on the lubrication theory, due to the need for comparison with wind tunnel test data to validate the simulation accuracy, 2D models were primarily used in previously numerical simulation researches on RWIV. In the wind tunnel test referenced in the past, the wind tunnel was relatively small, the cable was rigid. The cable was relatively short and the deformation of cable was ignored. Therefore, 2D models could be used to simulate the vibration of 3D rigid cable. This method is simple, computationally efficient. The simulated cable vibration results are also close to the results of 3D rigid cable wind tunnel tests. However, in this study, reference is made to the latest wind tunnel test of 3D flexible cable conducted by Gao et al. [38]. In this wind tunnel test, the length of the cable model was 8.31 m, and it exhibited multiple natural frequencies. Additionally, the wind tunnel was also larger than before, as shown in Fig. 21. Based on the wind tunnel test of 3D flexible cable, Gao et al. [38] found that the vibration mode, dominant frequency, and shape of the cable was varied with varying wind speeds during RWIV (Fig. 22). Above phenomenon was not found in previous wind tunnel tests of 3D rigid cable. Consequently, it is impossible to simulate the multiple mode, multiple frequency, and morphological variation of cable vibration under different wind speeds during RWIV through using the 2D simulation method. Moreover, in this paper, a 3D flexible cable model is developed by using ansys software to simulate the cable vibration, not only due to the reason of wind tunnel tests mentioned above. Matsumoto et al. [9,12,13]. Found that the axial flow of air and rivulets can also have a significant impact on the cable vibration. It is likely that this axial flow contributes to the variation in vibration mode and dominant frequency of the cables under varying wind speeds during RWIV. Therefore, to gain a deeper understanding of the RWIV phenomenon, it is essential to develop numerical simulation model for the gas, liquid, and solid phases to 3D models.

Fig. 21

View large Download slide

Three-dimensional rigid cable model of wind tunnel test

Fig. 22

View large Download slide

Three-dimensional flexible cable model of wind tunnel test by Gao et al

4 Conclusion

The model of 3D flexible cable established in ansys is used to simulate the cable vibration response. The aerodynamic forces are calculated by using comsol. The evolution of water film and cable lift are calculated by using matlab The cable lift imported into the cable model at each time-step can be updated by using the technology of ansys Restart. matlab is also used to control the start of comsol and ansys, achieving the transmission of information. As a result, the new numerical simulation method of RWIV of 3D flexible cable segment can be proposed. By using this method and considering the influence of cable deformation, the water film evolution, cable lift variation, and cable vibration are studied. Some conclusions can be drawn:

The evolution of water film at different positions on the cable surface is different. The water film thickness at the mid span of the cable is thicker than on both sides of the cable, and the amount of dropping water at the position of upper rivulet is greater.
By using the 3D flexible cable model in ansys to simulate the cable vibration response, the displacement of cable vibration is acquired at different positions on the cable surfface. The spatial shape of cable vibration at each moment is also obtained
The co-frequency resonance between the cable vibration and the water film evolution plays a significant role in RWIV.

Funding Data

National Natural Science Foundation of China Youth, Project “Failure Mechanism of Corrosion and Load Coupling, and Prediction of Deterioration Process of Welded Joints of Bare Weathering Steel Bridges” (Grant No. 52208194).
Tianjin Natural Science Foundation Youth, Project “Analysis of Local Corrosion of Concrete-Filled Steel Tube Truss Joints in Coastal Environments and Its Evolution Mechanism” (Grant No. 22JCQNJC01550).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

References

1.

Bosdogianni

,

A.

, and

Olivari

,

D.

,

1996

, “

Wind-and-Rain-Induced Oscillations of Cables of Stayed Bridges

,”

J. Wind Eng. Ind. Aerodyn.

,

64

(

2–3

), pp.

171

–

185

.10.1016/S0167-6105(96)00089-X

2.

Hikami

,

Y.

, and

Shiraishi

,

N.

,

1988

, “

Rain-Wind Induced Vibrations of Cables Stayed Bridges

,”

J. Wind Eng. Ind. Aerodyn.

,

29

(

1–3

), pp.

409

–

418

.10.1016/0167-6105(88)90179-1

3.

Geurts

,

C.

,

Vrouwenvelder

,

T.

,

van Staalduinen

,

P.

, and

Reusink

,

J.

,

1998

, “

Numerical Modeling of Rain-Wind-Induced Vibration: Erasmus Bridge, Rotterdam

,”

Struct. Eng. Int.

,

8

(

2

), pp.

129

–

135

.10.2749/101686698780489351

Google Scholar

Crossref

4.

Zuo

,

D.

,

Jones

,

N. P.

, and

Main

,

J. A.

,

2008

, “

Field Observation of Vortex and Rain-Wind-Induced Stay-Cable Vibrations in a Three-Dimensional Environment

,”

J. Wind Eng. Ind. Aerodyn.

,

96

(

6–7

), pp.

1124

–

1133

.10.1016/j.jweia.2007.06.046

5.

Ni

,

Y. Q.

,

Wang

,

X. Y.

,

Chen

,

Z. Q.

, and

Ko

,

J. M.

,

2007

, “

Field Observations of Rain-Wind-Induced Cable Vibration in Cable-Stayed Dongting Lake Bridge

,”

J. Wind Eng. Ind. Aerodyn.

,

95

(

5

), pp.

303

–

328

.10.1016/j.jweia.2006.07.001

Google Scholar

Crossref

6.

Ming

,

G.

,

Cijun

,

L.

, and

Qiang

,

L. G.

,

1998

, “

Rain-Wind Induced Vibration of Cables on Cable-Stayed Bridges and Its Control

,”

Shanghai J. Mech.

,

9

(

4

), pp.

281

–

288

.https://link.oversea.cnki.net/doi/10.15959/j.cnki.0254-0053.1998.04.001

7.

Main

,

J. A.

, and

Jones, N

,

P.

,

1999

, “

Full Scale Measurements of Stay Cable Vibration

,”

Proceeding of the 10th International Conference on Wind Engineering

, Copenhagen, Denmark, June 21–24, pp. 963–970.http://www.jmain.sent.com.user.fm/pubs/Main-Jones-ICWE10.pdf

Google Scholar

Crossref

8.

Matsumoto

,

M.

,

Yokoyama

,

K.

, and

Miyata

,

T.

,

1989

, “

Wind-Induced Cable Vibration of Cable-Stayed Bridges in Japan

,”

Proceedings of the Canada–Japan Workshop on Bridge Aerodynamics

, Ottawa, Canada, pp.

101

–

110

.

9.

Matsumoto

,

M.

,

Shiraishi

,

N.

,

Kitazawa

,

M.

,

Knisely

,

C.

,

Shirato

,

H.

,

Kim

,

Y.

, and

Tsujii

,

M.

,

1990

, “

Aerodynamic Behavior of Inclined Circular Cylinders-Cable Aerodynamics

,”

J. Wind Eng. Ind. Aerodyn.

,

33

(

1–2

), pp.

63

–

72

.10.1016/0167-6105(90)90021-4

10.

Matsumoto

,

M.

,

Shiraishi

,

N.

, and

Shirato

,

H.

,

1992

, “

Rain-Wind Induced Vibration of Cables of Cable-Stayed Bridges

,”

J. Wind Eng. Ind. Aerodyn.

,

43

(

1–3

), pp.

2011

–

2022

.10.1016/0167-6105(92)90628-N

11.

Matsumoto

,

M.

,

Saitoh

,

T.

,

Kitazawa

,

M.

,

Shirato

,

H.

, and

Nishizaki

,

T.

,

1995

, “

Response Characteristics of Rain-Wind Induced Vibration of Stay-Cables of Cable-Stayed Bridges

,”

J. Wind Eng. Ind. Aerodyn.

,

57

(

2–3

), pp.

323

–

333

.10.1016/0167-6105(95)00010-O

12.

Matsumoto

,

M.

,

Yagi

,

T.

,

Goto

,

M.

, and

Sakai

,

S.

,

2003

, “

Rain–Wind-Induced Vibration of Inclined Cables at Limited High Reduced Wind Velocity Region

,”

J. Wind Eng. Ind. Aerodyn.

,

91

(

1–2

), pp.

1

–

12

.10.1016/S0167-6105(02)00331-8

13.

Matsumoto

,

M.

,

Yagi

,

T.

,

Shigemura

,

Y.

, and

Tsushima

,

D.

,

2001

, “

Vortex-Induced Cable Vibration of Cable-Stayed Bridges at High Reduced Wind Velocity

,”

J. Wind Eng. Ind. Aerodyn.

,

89

(

7–8

), pp.

633

–

647

.10.1016/S0167-6105(01)00063-0

14.

Yamaguchi

,

H.

,

1990

, “

Analytical Study on Growth Mechanism of Rain Vibration of Cables

,”

J. Wind Eng. Ind. Aerodyn.

,

33

(

1–2

), pp.

73

–

80

.10.1016/0167-6105(90)90022-5

15.

Jing

,

H.

,

Xia

,

Y.

,

Li

,

H.

,

Xu

,

Y.

, and

Li

,

Y.

,

2015

, “

Study on the Role of Rivulet in Rain–Wind-Induced Cable Vibration Through Wind Tunnel Testing

,”

J. Fluids Struct.

,

59

(

10

), pp.

316

–

327

.10.1016/j.jfluidstructs.2015.09.008

16.

Li

,

Y.

,

Jing

,

H.

,

Xia

,

Y.

,

Xu

,

Y.

, and

Xiang

,

H.

,

2015

, “

Measurement of Rivulet Movement on Inclined Cables During Rain–Wind Induced Vibration

,”

Sens. Actuators A: Phys.

,

230

, pp.

17

–

24

.10.1016/j.sna.2015.03.040

Google Scholar

Crossref

17.

Jing

,

H.

,

Xia

,

Y.

,

Li

,

H.

,

Xu

,

Y.

, and

Li

,

Y.

,

2017

, “

Excitation Mechanism of Rain–Wind Induced Cable Vibration in a Wind Tunnel

,”

J. Fluids Struct.

,

68

, pp.

32

–

47

.10.1016/j.jfluidstructs.2016.10.006

Google Scholar

Crossref

18.

Peil

,

U.

, and

Nahrath

,

N.

,

2003

, “

Modeling of Rain-Wind Induced Vibration

,”

Wind Struct.

,

6

(

1

), pp.

41

–

52

.10.12989/was.2003.6.1.041

Google Scholar

Crossref

19.

Lv

,

Q.

, and

Gu

,

M.

,

2001

, “

The Analytical Model of Rain-Wind Vibration of Stay Cables

,”

Struct. Eng.

, 1, pp.

12

–

15

.https://link.oversea.cnki.net/doi/10.15935/j.cnki.jggcs.2001.01.003

20.

Gu

,

M.

,

Li

,

S. Y.

, and

Du

,

X. Q.

,

2007

, “

Testing Study on Wind Pressure Distributions of Stayed Cables With a Fixed Artificial Rivulet

,”

Acta Aerodyn. Sin.

,

25

(

2

), pp.

169

–

174

.http://kqdlxxb.xml-journal.net/article/id/9934

21.

Li

,

S. Y.

,

Gu

,

M.

, and

Chen

,

Z. Q.

,

2007

, “

Analytical Model for Rain-Wind-Induced Vibration of Three-Dimensional Continuous Stay Cable With Quasi-Moving Rivulet

,”

Eng. Mech.

,

24

(

6

), pp.

7

–

14

.https://www.researchgate.net/publication/279901576_Analytical_model_for_rain-wind-induced_vibration_of_three-dimensional_continuous_stay_cable_with_quasi-moving_rivulet

Google Scholar

Crossref

22.

Lemaitre

,

C.

,

Mahmud Alam

,

M.

,

Hémon

,

P.

,

de Langre

,

E.

, and

Zhou

,

Y.

,

2006

, “

Rainwater Rivulets on a Cable Subject to Wind

,”

C. R. Méc.

,

334

(

3

), pp.

158

–

163

.10.1016/j.crme.2006.01.005

Google Scholar

Crossref

23.

Lemaitre

,

C.

,

Hémon

,

P.

, and

Langre

,

E.

,

2007

, “

Thin Water Film Around a Cable Subject to Wind

,”

J. Wind Eng. Ind. Aerodyn.

,

95

(

9–11

), pp.

1259

–

1271

.10.1016/j.jweia.2007.02.007

24.

Lemaitre

,

C.

,

Langre

,

E.

, and

Hémon

,

P.

,

2010

, “

Rainwater Rivulets Running on a Stay Cable Subject to Wind

,”

Eur. J. Mech. B/Fluids

,

29

(

4

), pp.

251

–

258

.10.1016/j.euromechflu.2010.02.007

Google Scholar

Crossref

25.

Robertson

,

A. C.

,

Taylor

,

I. J.

,

Wilson

,

S. K.

,

Duffy

,

B. R.

, and

Sullivan

,

J. M.

,

2010

, “

Numerical Simulation of Rivulet Evolution on a Horizontal Cable Subject to an External Aerodynamic Field

,”

J. Fluids Struct.

,

26

(

1

), pp.

50

–

73

.10.1016/j.jfluidstructs.2009.09.003

Google Scholar

Crossref

26.

Taylor

,

I. J.

, and

Robertson

,

A. C.

,

2011

, “

Numerical Simulation of the Airflow–Rivulet Interaction Associated With the Rain-Wind Induced Vibration Phenomenon

,”

J. Wind Eng. Ind. Aerodyn.

,

99

(

9

), pp.

931

–

944

.10.1016/j.jweia.2011.03.012

Google Scholar

Crossref

27.

Taylor

,

I. J.

, and

Robertson

,

A. C.

,

2015

, “

Numerical Investigation of the Coupled Interaction Between an Unsteady Aerodynamic Flow Field and a Water Film Coating on a Circular Cylinder

,”

J. Fluids Struct.

,

54

, pp.

312

–

331

.10.1016/j.jfluidstructs.2014.11.008

Google Scholar

Crossref

28.

Xu

,

L. S.

,

Ge

,

Y. J.

, and

Zhao

,

L.

,

2011

, “

Experimental Study of Rain and Wind-Induced Vibration of Stay Cables Using a High-Precision Rain Simulation System

,”

China Civ. Eng. J.

,

44

(

5

), pp.

86

–

93

.https://link.oversea.cnki.net/doi/10.15951/j.tmgcxb.2011.05.005

29.

Jihong

,

B.

,

Jian

,

W.

,

Peng

,

L.

, and

Chun

,

B.

,

2014

, “

Variation of Water Film Morphology and Aerodynamic Force of Stay Cable

,”

Natural Sci. Eng. Technol.

,

47

(

6

), pp.

479

–

490

.

30.

Jihong

,

B.

,

Jian

,

W.

,

Qian

,

S.

, and

Peng

,

L.

,

2014

, “

2-D Coupling Between Water Film Morophology and Rain-Wind Induced Vibration of Cable

,”

Eng. Mech.

,

31

(

7

), pp.

54

–

60 + 77

.https://link.oversea.cnki.net/doi/10.6052/j.issn.1000-4750.2012.11.0909

31.

Bi

,

J. H.

,

Wang

,

J.

,

Shao

,

Q.

,

Lu

,

P.

,

Guan

,

J.

, and

Li

,

Q. B.

,

2013

, “

2D Numerical Analysis on Evolution of Water Film and Cable Vibration Response Subject to Wind and Rain

,”

J. Wind Eng. Ind. Aerodyn.

,

121

, pp.

49

–

59

.10.1016/j.jweia.2013.07.018

Google Scholar

Crossref

32.

Jian

,

W.

,

Peng

,

L.

,

Jihong

,

B.

, et al.,

2016

, “

Application of Slip Theory in the Study of Two-Degree of Freedom Model of Wind and Rain Induced Vibration of Stayed Cables

,”

J. Vib. Eng.

,

29

(

004

), pp.

737

–

745

.https://link.oversea.cnki.net/doi/10.16385/j.cnki.issn.1004-4523.2016.04.022

33.

Wang

,

J.

,

Lu

,

P.

,

Bi

,

J. H.

,

Guan

,

J.

, and

Qiao

,

H. Y.

,

2016

, “

Three-Phase Coupled Modelling Research on Rain–Wind Induced Vibration of Stay Cable Based on Lubrication Theory

,”

J. Fluids Struct.

,

63

, pp.

16

–

39

.10.1016/j.jfluidstructs.2016.02.008

Google Scholar

Crossref

34.

Jihong

,

B.

,

Ji

,

W.

,

Jian

,

G.

, and

Jian

,

W.

,

2017

, “

Numerical Simulation of Rain-Wind Induced Vibration of Stay Cables and Its Mechanism Study

,”

J. Vib. Shock.

,

36

(

11

), pp.

111

–

117

.https://link.oversea.cnki.net/doi/10.13465/j.cnki.jvs.2017.11.017

35.

Bi

,

J. H.

,

Guan

,

J.

,

Wang

,

J.

,

Lu

,

P.

,

Qiao

,

H. Y.

, and

Wu

,

J.

,

2018

, “

3D Numerical Analysis on Wind and Rain Induced Oscillations of Water Film on Cable Surface

,”

J. Wind Eng. Ind. Aerodyn.

,

176

, pp.

273

–

289

.10.1016/j.jweia.2018.03.026

Google Scholar

Crossref

36.

Jian

,

W.

,

Jihong

,

B.

,

Jian

,

G.

,

Haoyue

,

Q.

,

Qian

,

S.

,

Yan

,

Z.

, and

Qinghai

,

G.

,

2020

, “

Three-Dimensional Numerical Simulation on Rivulet Movement of Rain-Wind Induced Vibration of Stay Cable

,”

J. Vib. Eng.

,

33

(

3

), pp.

559

–

569

.https://link.oversea.cnki.net/doi/10.16385/j.cnki.issn.1004-4523.2020.03.015

37.

Jian

,

W.

,

Jihong

,

B.

, and

Peng

,

L.

,

2020

, “

3-D Numerical Simulation for bi-Directional Coupling Between Waterline Motion and Cable's Rain-Wind Induced Vibration

,”

J. Vib. Shock

,

39

(

3

), pp.

8

–

15

.https://link.oversea.cnki.net/doi/10.13465/j.cnki.jvs.2020.03.002

38.

Gao

,

D.

,

Chen

,

W.

,

Eloy

,

C.

, and

Li

,

H.

,

2018

, “

Multi-Mode Responses, Rivulet Dynamics, Flow Structures and Mechanism of Rain-Wind Induced Vibrations of a Flexible Cable

,”

J. Fluids Struct.

,

82

, pp.

154

–

172

.10.1016/j.jfluidstructs.2018.06.017

Google Scholar

Crossref

39.

Menter

,

F. R.

,

1994

, “

Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications

,”

AIAA J.

,

32

(

8

), pp.

1598

–

1605

.10.2514/3.12149

Google Scholar

Crossref

40.

Norberg

,

C.

,

2003

, “

Fluctuating Lift on a Circular Cylinder: Review and New Measurements

,”

J. Fluids Struct.

,

17

(

1

), pp.

57

–

96

.10.1016/S0889-9746(02)00099-3

Google Scholar

Crossref

41.

Sun

,

W. F.

,

2011

, “

The Mechanism Study of Rain-Wind Induced Vibration of Stay Cables Based on the Forced Vibration Device

,”

Hunan University

,

Changsha

.

42.

Li

,

D.

,

2013

, “

Study on the Theoretical Models of Rain-Wind-Induced Vibration of Cable and Analysis on Vibration Characteristics

,”

Hunan University

,

Changsha

.

43.

Li

,

H.

,

Chen

,

W. L.

,

Xu

,

F.

,

Li

,

F. C.

, and

Ou

,

J. P.

,

2010

, “

A Numerical and Experimental Hybrid Approach for the Investigation of Aerodynamic Forces on Stay Cables Suffering From Rain-Wind Induced Vibration

,”

J. Fluids Struct.

,

26

(

7–8

), pp.

1195

–

1215

.10.1016/j.jfluidstructs.2010.06.006

Numerical Simulation Method for the Rain-Wind Induced Vibration of the Three-Dimensional Flexible Stay Cable

Abstract

1 Introduction

2 Model

2.1 Evolution Equation of Water Film.

2.2 Equation of the Cable Lift.

2.3 The Computational Fluid Dynamics Model.

2.4 Three-Dimensional Flexible Stay Cable Model in ansys.

2.5 Numerical Solution Flow and Basic Parameters.

2.5.1 The Flow of Numerical Solutions.

2.5.2 Basic Parameter.

3 Numerical Simulation Result

3.1 The Evolution of Water Film.

3.1.1 The Formation of the Rivulet.

3.1.2 Evolution of Water Film at Specific Positions.

3.1.3 Circumferential Evolution Analysis of the Rivulet.

3.1.4 Axial Evolution Analysis of the Rivulet.

3.2 Variation of the Cable Lift.

3.3 Response of the Cable Vibration.

3.4 Discussion.

4 Conclusion

Funding Data

Data Availability Statement

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

$γ$	Surface tension coefficient of water in air	$ρ$	Density of water
P_g	Air pressure on the surface of the water film	g	Acceleration of gravity
U_N	Vertical component of wind speed	ρ_g	Air density
$t_{g θ}$	Circumferential vectors along the cable at the junction of water film and air	$C_{f Z} = \frac{t_{Z}}{\frac{1}{2} ρ_{g} U_{N}^{2}}$	Wind friction coefficient along the cable axis
$t_{g Z}$	Axial tangential vectors along the cable at the junction of water film and air	$C_{f θ} = \frac{t_{θ}}{\frac{1}{2} ρ_{g} U_{N}^{2}}$	Wind friction coefficient along the circumferential of the cable
K	Curvature of the surface of the water film	$C_{p} = \frac{P_{g}}{\frac{1}{2} ρ_{g} U_{N}^{2}}$	Wind pressure coefficient
μ	Water's dynamic viscosity coefficient	$\ddot{y}$	Cable acceleration
g_N	Gravitational acceleration component in the plane	g_Z	Axial acceleration component of gravity
R	Cable radius	h	Water film thickness

Numerical Simulation Method for the Rain-Wind Induced Vibration of the Three-Dimensional Flexible Stay Cable

Abstract

1 Introduction

2 Model

2.1 Evolution Equation of Water Film.

2.2 Equation of the Cable Lift.

2.3 The Computational Fluid Dynamics Model.

2.4 Three-Dimensional Flexible Stay Cable Model in ansys.

2.5 Numerical Solution Flow and Basic Parameters.

2.5.1 The Flow of Numerical Solutions.

2.5.2 Basic Parameter.

3 Numerical Simulation Result

3.1 The Evolution of Water Film.

3.1.1 The Formation of the Rivulet.

3.1.2 Evolution of Water Film at Specific Positions.

3.1.3 Circumferential Evolution Analysis of the Rivulet.

3.1.4 Axial Evolution Analysis of the Rivulet.

3.2 Variation of the Cable Lift.

3.3 Response of the Cable Vibration.

3.4 Discussion.

4 Conclusion

Funding Data

Data Availability Statement

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only