Aerodynamic and Vibration Characteristics of Iced Power Transmission Conductors in a Nonuniform Wind Field Based on Unsteady Theory

Abstract

To study the aerodynamic and vibration characteristics of iced conductors under the influence of wind fluctuations, a harmonic superposition method is used to simulate nonuniform wind speeds. A user-defined function is written on the basis of the secondary development function of the Fluent 2021 R1 software to determine the displacement and velocity of the conductor at each time step, and a two-way fluid–structure interaction (FSI) numerical simulation of an iced conductor under a nonuniform wind field is performed via an overset mesh method. In the analysis, the aerodynamic coefficients and galloping characteristics of iced conductors under different degrees of freedom (DOFs) are investigated by considering different combinations of quasi-steady theory, unsteady theory, a uniform wind field, and a nonuniform wind field. The results show that in a nonuniform wind field, the mean, standard deviation (SD), and peak values of the drag and torsion coefficients of the conductors calculated via unsteady theory are significantly larger than those calculated via quasi-steady theory, indicating that the obtained aerodynamic coefficients of the latter (the mean values are typically used) conceal the characteristics of the iced conductors in an actual wind environment and ignore the adverse effects of the variability.

1. Introduction

The galloping of iced power transmission conductors is a low-frequency, large-amplitude self-excited vibration phenomenon caused by irregular ice coatings on conductors [1,2], and alleviating the galloping of iced transmission conductors is particularly complicated because of their highly nonlinear characteristics and many influencing factors. Since the 1930s, researchers in China and abroad have been studying this problem without interruption, but a reasonable solution has still not been obtained, and line damage accidents, such as phase-to-phase flashovers, power outages due to tripping, broken conductor strands (ground), damaged fittings and insulators, loosened and falling poles and tower bolts, and line breaks and tower collapses, still occur frequently [3,4] and pose a serious threat to the safe operation of power systems. Therefore, the in-depth study of galloping iced transmission conductors still has important practical significance.

The widely recognized galloping mechanisms of iced transmission conductors include Den Hartog’s vertical galloping mechanism and Nigol’s torsional galloping mechanism [5,6]. In the vertical galloping mechanism, the aerodynamic force exerted on the iced conductor under the action of the wind is divided into a drag force along the wind direction and a lift force in the crosswind direction, and when the sum of the slopes of the lift coefficient curve and drag coefficient curve is negative, the system can experience negative aerodynamic damping, causing the iced conductor to vibrate in the crosswind direction. The torsional galloping mechanism infers that the torsion of the conductor itself could increase the negative damping of the iced conductor vibration in the crosswind direction; therefore, when the damping of torsional vibrations of the iced transmission conductor is less than zero at a 0° angle of attack, self-excited torsional vibration occurs. These two mechanisms infer that when the negative aerodynamic damping of a conductor is greater than the structural damping of the conductor itself, galloping can ensue. The galloping determination criteria are based on the aerodynamic coefficients of the conductor; therefore, the aerodynamic characteristics of the conductor are the basis for studying conductor galloping.

Previous studies [7,8,9,10] have shown that factors such as ice thickness, wind speed, conductor diameter, and wind attack angle affect the aerodynamic characteristics of iced conductors. However, current studies on the aerodynamic characteristics of conductors have been mostly based on quasi-steady theory [11,12], in which the static aerodynamic coefficients of the conductor are calculated without considering the interaction between the flow field and the conductor and then applied to simulation analysis of the galloping response of the conductor. For example, Ishihara and Oka [13] used an LES turbulence model to numerically simulate a single conductor and a quad-bundle conductor with crescent-shaped iced sections, and the influence of the wake effect of conductors on aerodynamic characteristics was analyzed. On the basis of quasi-steady theory, Wu et al. [14] simulated the vibration behavior of a double-bundle conductor in a two-dimensional (2D) flow field to obtain the aerodynamic characteristics of the conductor, which were subsequently applied to a finite element model to analyze conductor galloping. Lou et al. [15] conducted numerical simulations on an 8-bundled conductor with D-shaped icing; using the Galerkin method, they systematically studied the impact of different wind attack angles and speeds on the galloping amplitudes in the vertical, horizontal, and torsional directions. The full range of wind attack angles was divided into five galloping regions of varying severity. Liao et al. [16] used the Spalart–Allmaras (S-A) model to simulate and analyze a conductor with crescent-shaped iced sections; the effects of the ice thickness, wind speed, and wind attack angle on the aerodynamic coefficient of the conductor were systematically studied, and a wind attack angle interval suitable for conductor galloping with crescent-shaped iced sections was analyzed in combination with the galloping mechanism. These studies neglected the coupling between the conductor and the wind in their aerodynamic analyses. However, owing to the influence of conductor vibration, the aerodynamic force of a conductor can constantly change with changes in the wind speed, conductor position, and conductor trajectory; therefore, several researchers have studied the dynamic aerodynamic coefficients of conductors. Li et al. [17] used static and dynamic wind tunnel tests to study conductors with crescent- and sector-shaped iced sections and reported that the static aerodynamic coefficient roughly reflects the dynamic aerodynamic trend of the conductor, but the magnitude of the negative aerodynamic damping and attack angle obtained via the dynamic test method may differ from that of the static test. On the basis of an elliptic cylinder wind tunnel test, Ma et al. [18] reported that the quasi-steady theory was not suitable for predicting conductor galloping at the critical Reynolds number. Yang et al. [19] performed a 2D numerical simulation of the aerodynamic characteristics of a conductor considering fluid–structure interaction (FSI), compared the results with those of quasi-steady theory, and discussed the applicability of the two theories. Liu et al. [20] studied the aerodynamic characteristics of 4-bundled conductors through aeroelastic model wind tunnel tests and reported that conductor drag and tension increase with increasing turbulence intensity. Azzi et al. [21] used a 1:50 scale model and wind tunnel tests to study transmission line systems; by analyzing the response of multi-span transmission line systems under different wind speeds and directions, they reported that coupling effects significantly alter aerodynamic damping, and turbulence intensity variation improves response estimates. Cai et al. [22] conducted a wind tunnel test study of the dynamic aerodynamic coefficients of a quad-bundle conductor and compared the results with those of the traditional method; their results revealed that the dynamic aerodynamic coefficient was greater than the static aerodynamic coefficient, and their analysis also validated that using quasi-steady theory to analyze the aerodynamic forces of iced conductors was not entirely suitable. Therefore, when performing aerodynamic analysis on an iced conductor, it is necessary to consider the coupling between the wind and the conductor to simulate the forces on the conductor more accurately.

Obviously, natural wind introduces obvious fluctuations [23,24,25] that have a nonnegligible effect on the aerodynamic characteristics of a conductor. However, this phenomenon has not been sufficiently considered in previous studies on iced conductor galloping. There has been a lack of sufficient computing resources, and since galloping causes are very complicated, researchers have often used simplified approaches when analyzing the effects of the most important factors on conductor galloping, e.g., by considering only the response of a conductor under the action of a uniform flow field [10,26]. In recent years, with the development of computing technology, numerical simulations based on massive parallel computing have gradually become the mainstream way to study iced conductor galloping. Furthermore, recent studies have begun to explore more complex flow conditions. For example, Brusco et al. [27] investigated the transient aerodynamics of a square cylinder under downburst-like accelerating flows in a multiple-fan wind tunnel. Lunghi et al. [28] examined the influence of inflow acceleration on the aerodynamic characteristics of a square cylinder through numerical simulations. These studies highlight the importance of accounting for higher temporal variability in wind speed to accurately capture the aerodynamic behavior under nonstationary conditions. Kim and Sohn [29] used ANSYS Fluent software to simulate the aerodynamic coefficients of iced conductors with crescent- and conical-shaped iced sections, performed a galloping analysis under a nonuniform wind field, and concluded that a conical-shaped iced section had a greater impact on galloping than a crescent-shaped iced section in a nonuniform wind field. However, Kim and Sohn [29] did not perform flow field analysis under a nonuniform wind field, so the obtained aerodynamic coefficients were derived under a uniform wind field. Wei et al. [30] set the galloping trajectory of power transmission conductors to an elliptical motion and simulated the movement of iced conductors in a nonuniform wind field; however, they did not consider the effect of iced conductor vibration on the flow field. Liu et al. [31] studied the stability and response analysis methods of iced conductors under the action of uniform wind and turbulent wind, and their comparison of calculation examples verified that wind pulsations have a certain effect on the galloping amplitude of iced conductors. On the basis of a finite element model of conductors with crescent-shaped iced sections, Chen et al. [32] studied the galloping of an iced conductor under the actions of steady, unsteady, and random wind fields, and their results revealed that the galloping amplitude of a conductor under the action of a random wind field was much greater than that under the action of a steady wind field. Therefore, the wind field has a great effect on conductor galloping. To simulate the aerodynamic characteristics and galloping responses of iced conductors more accurately, the wind fluctuation characteristics should be considered.

In summary, owing to the limitations of experimental and computational technologies, most current studies still adopt quasi-steady theory and rarely systematically consider the dynamic influence of conductor vibrations on the flow field. Moreover, existing numerical studies on the aerodynamics and galloping of iced transmission lines based on fluid–structure interactions (FSIs) are mostly conducted under uniform wind fields and fail to fully account for the fluctuating characteristics of airflow, which significantly differ from the complexity of actual wind fields.

To address the above issues, this paper innovatively adopts the unsteady theory and utilizes the secondary development functionality of ANSYS Fluent software in combination with moving mesh technology to conduct FSI simulations of iced conductors in nonuniform wind fields on a supercomputing platform. By employing the harmonic superposition method to generate nonuniform wind speeds, this study is the first to apply unsteady theory to the aerodynamic characteristics of iced conductors in nonuniform wind fields. A comparative analysis with results derived from quasi-steady theory under uniform wind fields is conducted, revealing the aerodynamic coefficient characteristics of iced conductors in nonuniform wind fields under unsteady conditions. The findings of this research provide theoretical support for a comprehensive understanding of the galloping behavior of iced conductors in nonuniform wind fields and offer significant guidance for developing more scientific and effective galloping control strategies.

The remainder of this paper is organized as follows: Section 2 introduces the theoretical framework and analytical model, including the governing equations of the fluid–structure interaction (FSI) problem and the numerical implementation methods. Section 3 presents the harmonic superposition method used to simulate nonuniform wind fields and describes its specific implementation in ANSYS Fluent. Section 4 analyzes the aerodynamic characteristics and galloping behavior of iced conductors under different wind field conditions and compares the results obtained via quasi-steady and unsteady theories. Finally, Section 5 summarizes the key findings and discusses their significance for the development of galloping control strategies.

2. Theoretical and Analytical Models

Since the analysis of an iced conductor via unsteady theory is a two-way FSI problem in computational fluid dynamics, this paper uses a mass–spring-damping system to simulate the movement of the iced conductor in a flow field, dynamically calculates the solid deformation field by analyzing the vibrations of the conductor in different directions, and uses ANSYS Fluent 2021 R1 software and its secondary development function to realize bidirectional data transmission.

2.1. Governing Equations for the Fluid Domain

The governing equations for the fluid domain are described by the continuity equation and the N–S equations for a 2D incompressible viscous fluid.

Continuity equation:

$${\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}=0$$

(1)

Momentum equations:

$${\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}={\displaystyle \frac{\mu }{\rho }}({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}})-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x}}$$

(2)

$${\displaystyle \frac{\partial v}{\partial t}}+u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}={\displaystyle \frac{\mu }{\rho }}({\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}v}{\partial y^2}})-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x}}$$

(3)

where ρ is the fluid density, μ is the dynamic viscosity coefficient, p is the fluid pressure, and u and v are the fluid velocities in the x and y directions, respectively.

For the numerical simulation of turbulence, the Reynolds-averaged Navier–Stokes (RANS) method was selected. RANS is a method based on time-averaged turbulence decomposition, which divides the flow field variables into mean and fluctuating components. By solving the N–S equations and performing temporal averaging on the instantaneous turbulence motion terms, RANS reduces computational requirements while capturing all turbulent motion.

The turbulence model adopted is the $SST k-\omega$ model, which shares the advantages of the standard $k-\omega$ model, including better accuracy in predicting results in the near-wall region and separation flows, as well as streamlined computations. However, unlike the standard $k-\omega$ model, which is overly sensitive to boundary conditions, the $SST k-\omega$ model can simulate flow separation more effectively.

Introducing the Reynolds-averaged flow field allows physical quantities in the flow field to be expressed as the sum of their mean and fluctuating components, $u_i=\overline{u_i}+u^{\prime },p=\overline{p}+p^{\prime }$. The motion equation for Reynolds-averaged flow can then be expressed as follows:

$${\displaystyle \frac{\partial }{\partial t}}(\overline{u_i})+{\displaystyle \frac{\partial }{\partial x_j}}(\overline{u_i}\overline{u_j})=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left({\displaystyle \frac{\mu }{\rho }}{\displaystyle \frac{\partial \overline{u_i}}{\partial x_j}}-\overline{u_i^{\prime }}\overline{u_j^{\prime }}\right)$$

(4)

The overlined physical variables represent time-averaged variables. In Equation (4), each term corresponds to fluid acceleration, viscous stress, pressure, or turbulence.

The physical phenomenon of turbulence includes nonlinear characteristics, leading to the presence of turbulence forces, which express the momentum dispersion effect of the average flow under the action of turbulence. The turbulence term includes the newly introduced Reynolds stresses, which add unknown variables to the governing equations (N–S equations) and render them unsolvable. To close motion Equation (4), the Boussinesq hypothesis is needed. For incompressible fluids, the Reynolds stress term is expressed as follows:

$$-\rho \overline{u_i^{\prime }}\overline{u_j^{\prime }}={\mu }_t\left({\displaystyle \frac{\partial \overline{u_i}}{\partial x_j}}+{\displaystyle \frac{\partial \overline{u_j}}{\partial x_i}}\right)-{\displaystyle \frac{2}{3}}(\rho k+{\mu }_t{\displaystyle \frac{\partial u_i}{\partial x_i}}){\delta }_{ij}$$

(5)

Here, ${\delta }_{ij}$ is the Kronecker delta (${\delta }_{ij}=1$ when $i=j$; otherwise, ${\delta }_{ij}=0$), and $k$ represents the turbulent kinetic energy:

$$k=={\displaystyle \frac{1}{2}}(\overline{u_1^{\prime 2}}+\overline{u_2^{\prime 2}}+\overline{u_3^{\prime 2}})$$

(6)

After the Boussinesq hypothesis is introduced, the key is to determine the turbulent viscosity to close the equations. This leads to different turbulence models. The $SST k-\omega$ model determines the turbulent viscosity (via the turbulent kinetic energy and specific dissipation rate) through two partial differential equations. The governing equations are as follows:

$${\displaystyle \frac{\partial (\rho k)}{\partial t}}+{\displaystyle \frac{\partial (\rho k{u}_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}\left(T_k{\displaystyle \frac{\partial k}{\partial x_j}}\right)+G_k-Y_k+S_k$$

(7)

$${\displaystyle \frac{\partial (\rho \omega )}{\partial t}}+{\displaystyle \frac{\partial (\rho \omega u_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}\left(T_{\omega }{\displaystyle \frac{\partial \omega }{\partial x_j}}\right)+G_{\omega }-Y_{\omega }+D_{\omega }+S_{\omega }$$

(8)

In these equations:

$G_k,G_{\omega }$: Turbulent kinetic energy and $\omega$ generation terms caused by mean velocity gradients.

$T_k,T_{\omega }$: Diffusion rates of $k$ and $\omega$.

$Y_k,Y_{\omega }$: Dissipation rates of $k$ and $\omega$ caused by turbulence.

$S_k,S_{\omega }$: User-defined source terms.

$D_{\omega }$: Cross-diffusion term.

2.2. Discretization of the Governing Equations

This study employs the finite volume method (FVM) to discretize the computational domain. Using Fluent 2021 R1 software, the computational domain is divided into a finite number of small control volumes. By integrating the governing equations over each control volume, the equations are discretized into integral equations centered at the grid nodes of the control volumes. Furthermore, the Gauss divergence theorem and difference schemes are applied to spatially discretize the integral equations. The selection of an appropriate discretization scheme plays a critical role in improving the convergence speed and ensuring solution accuracy. Compared with second-order schemes, first-order schemes converge more slowly and offer lower accuracy. Therefore, second-order schemes are generally preferred. In this study, the pressure term is discretized via a second-order scheme, whereas the turbulence kinetic energy, momentum, and specific dissipation rate are discretized via a second-order upwind scheme.

2.3. Governing Equations for the Motion of Iced Conductors

When performing 2D aerodynamic analysis on an iced conductor, the model is usually simplified to an oscillator model in a flow field, and the aerodynamic coefficients of the conductor are obtained by determining the flow field force [33,34,35]. Figure 1 shows the calculation model for an iced conductor considering different degrees of freedom (DOFs). Note: 1-DOF indicates the crosswind vibration of the conductor; 3-DOFs indicate the coupled vibration in the crosswind direction, along-wind direction, and torsional direction.

In the past, when studying iced conductors via quasi-steady theory, the unidirectional FSI model shown in Figure 1a was generally used; the position of the conductor was assumed to be fixed, the motion of the conductor was not considered, and after setting the boundary conditions in the software, the aerodynamic coefficient of the conductor was obtained.

On the other hand, when unsteady theory is used for calculations, the motion of the conductor needs to be considered. Figure 1b,c show calculation models with 1-DOF and 3-DOFs, respectively. The motion equation of the 1-DOF oscillator model is shown in Equation (10). The motion equations of the 3-DOFs oscillator model are shown in Equations (9) and (10):

$$m\ddot{x}+2\xi {\omega }_{0}m\dot{x}+m{{\omega }_0}^{2}x=F_x$$

(9)

$$m\ddot{y}+2\xi {\omega }_{0}m\dot{y}+m{{\omega }_0}^{2}y=F_y$$

(10)

$$I_z\ddot{\theta }+2\xi {\omega }_0{I}_z\dot{\theta }+I_z{{\omega }_0}^2\theta =F_{\mathrm{z}}$$

(11)

where m and I_z are the mass and torsional stiffness of the system, respectively; x, y, and θ are the displacements in the horizontal, lateral, and torsional directions, respectively; ξ and ω₀ are the damping ratio and natural circular frequency of the system, respectively; and F_x, F_y, and F_z are the horizontal, lateral, and torsional forces on the system, respectively.

The conversion relationships between the aerodynamic coefficient and the aerodynamic force of the conductor are as follows:

$$F_d=0.5\rho u^{2}D{C}_d$$

(12)

$$F_l=0.5\rho u^{2}D{C}_l$$

(13)

$$F_m=0.5\rho u^2{D}^2{C}_m$$

(14)

where ρ is the air density; u is the relative wind speed; D is the conductor diameter; F_d, F_l, and F_m are the lift force, drag force, and torsion force, respectively; and C_d, C_l, and C_m are the lift coefficient, drag coefficient, and torsion coefficient, respectively.

2.4. Implementing Calculations Under Unsteady Theory

When performing 2D aerodynamic analysis on an iced conductor, the model is usually simplified to an oscillator model in a flow field, and the aerodynamic coefficients of the conductor are obtained by determining the flow field force. Overset mesh and moving mesh technologies are used to simulate the two-way FSI numerically between the conductor and the flow field. The fluid-governing equations are solved numerically to obtain the flow field information and the fluid force on the conductor surface. The fluid force acts on the iced conductor. The fourth-order Runge–Kutta method is used to solve the motion governing equations (Equations (9)–(11)) of the conductor to obtain the dynamic response (velocity and displacement) of the conductor in the x, y, and θ directions. Moving mesh technology is used to return the dynamic response of the iced conductor to the mesh system to update the mesh position. Moreover, an overset mesh is used to prevent mesh distortion during movement. When the flow field is stabilized, the results of the previous step are used as the start of the next step to calculate the response at the next time step, and by repeating this calculation, the dynamic response of the conductor at each time step can be obtained to realize a two-way FSI numerical simulation of the conductor. The specific iterative calculation process of the fourth-order Runge–Kutta method is as follows:

The force equilibrium equation acting on the system is as follows:

$$F_x(t)+F_y(t)+F_z(t)=P(t)$$

(15)

The equilibrium equation should be satisfied at time t + Δt:

$$F_x(t+\Delta t)+F_y(t+\Delta t)+F_z(t+\Delta t)=P(t+\Delta t)$$

(16)

where the incremental force can be expressed as follows:

$$\Delta F_x=F_x(t+\Delta t)-F_x(t)=m\Delta \ddot{y}(t)$$

(17)

$$\Delta F_y=F_y(t+\Delta t)-F_y(t)=c(t)\Delta \dot{y}(t)$$

(18)

$$\Delta F_z=F_z(t+\Delta t)-F_z(t)=k(t)\Delta y(t)$$

(19)

$$\Delta P(t)=P(t+\Delta t)-P(t)$$

(20)

The final increment equilibrium equation is as follows:

$$m\Delta \ddot{x}(t)+c(t)\Delta \dot{y}(t)+k(t)\Delta y(t)=\Delta P(t)$$

(21)

Using discretization, the speed increment and displacement increment at time t_n+1 can be obtained as follows [36,37,38]:

$$\Delta v_{(t_{n+1})}=\Delta v_{(t_n)}+\Delta t/6\times {(\mathrm{Q}}_1{+2\mathrm{Q}}_2{+2\mathrm{Q}}_3{+\mathrm{Q}}_4)$$

(22)

$$\Delta y(t_{n+1})=\Delta y(t_n)+\Delta v_{(t_n)}+{(\Delta t)}^2/6\times {(\mathrm{Q}}_1{+\mathrm{Q}}_2{+\mathrm{Q}}_3)$$

(23)

$$Q_1={\displaystyle \frac{\Delta P\left(t\right)}{m}}-c\left(t\right)\Delta v\left(t_n\right)-k\left(t\right)\Delta y\left(t_n\right)$$

(24)

$$Q_2={\displaystyle \frac{\Delta P_y\left(t\right)}{m}}-c\left(t\right)\left(\Delta v_{\left(t_0\right)}+{\displaystyle \frac{\Delta t}{2}}\cdot Q_1\right)-k\left(t\right)\left(\Delta y_{\left(t_n\right)}+{\displaystyle \frac{\Delta t}{2}}\cdot \Delta v\left(t_n\right)\right)$$

(25)

$$Q_3={\displaystyle \frac{\Delta P_y\left(t\right)}{m}}-c\left(t\right)\left(\Delta v_{\left(t_n\right)}+{\displaystyle \frac{\Delta t}{2}}\cdot Q_2\right)-k\left(t\right)\left(\Delta y_{\left(t_n\right)}+{\displaystyle \frac{\Delta t}{2}}\cdot \Delta v\left(t_n\right)+{\displaystyle \frac{{\left(\Delta t\right)}^2}{4}}\cdot Q_1\right)$$

(26)

$$Q_4={\displaystyle \frac{\Delta P_y\left(t\right)}{m}}-c\left(t\right)\left(\Delta v_{\left(t_n\right)}+{\displaystyle \frac{\Delta t}{2}}\right)\cdot Q_3-k\left(t\right)\left(\Delta y_{\left(t_n\right)}+\Delta t\cdot \Delta v_{\left(t_n\right)}+{\displaystyle \frac{{\left(\Delta t\right)}^2}{2}}\cdot Q_2\right)$$

(27)

The calculation process in ANSYS Fluent 2021 R1 software is implemented as follows. First, the model information is input, and boundary conditions, such as the wind speed, air density, and moving mesh, are set. During each time step, Fluent performs flow field calculations to obtain flow field-related parameters. The UDF program transfers the flow field-related parameters to the solid deformation field; then, the fourth-order Runge–Kutta method is used to calculate the displacement, velocity, and other information of the structure; and subsequently, the parameters are transmitted back to the flow field through the DEFINE_CG_MOTION macro to update the mesh. After the field has stabilized, the calculation result of the previous step is used as the initial value of the next step, and the loop is repeated until the termination number of the steps is reached. Figure 2 shows the specific calculation process.

2.5. Model Establishment

A 2D model is created by using a JL/G1A/400-35 conductor commonly used in 500 kV transmission lines as an example. This conductor is composed of inner steel cores and outer aluminum wires and has a cross-sectional area of 425.24 mm² and an outer diameter of 26.8 mm. A crescent-shaped iced section that is most likely to cause conductor galloping is selected [38], and the model cross-section is a combination of a semicircle and a semiellipse considering the twist characteristics. To fully simulate the development of turbulent flow while simultaneously meeting the constriction requirement, a rectangular flow field domain is used, and the cross-sections of the conductor and the calculation domain are shown in Figure 3. D represents the conductor diameter, and H represents the ice thickness.

2.6. Verification of Model Correctness and Mesh Independence

Meshing is the basis for flow field analysis, and a suitable mesh should meet the requirements of calculation accuracy and efficiency. To simulate the structural characteristics of conductors more accurately while ensuring mesh quality and calculation accuracy, a hybrid triangular and quadrilateral mesh is chosen in this study. The Y⁺ value is used to determine the near-wall mesh size, and the coarse mesh, medium mesh, and fine mesh forms are used to verify the effect of the mesh size on the calculation results. The specific forms of the foreground grid and background grid are shown in Figure 4.

To verify the correctness of the model established in this paper, a simulation analysis was performed using the same research object and working conditions as the wind tunnel test in Zdero and Turan [39]. Considering the twisting characteristics of the crescent-shaped iced section, the conductor has a diameter D of 30.5 mm, and the ice thicknesses are 1.1D and 1.4D. The wind attack angle is 0°, the wind speed is 5–20 m/s, and the interval between each calculation condition is 2.5 m/s.

Table 1. Selection of working conditions for the validation model.

Conductor Diameter (mm)	Ice Thickness (mm)	Wind Attack Angle (°)	Wind Speed (m/s)
30.5	1.1D, 1.4D	0	5, 7.5, 10, 12.5, 15, 17.5, 20

and

Table 2. Mesh parameters.

Mesh	Y+	Near-Wall Grid Size (mm)	Grid Size of Background (mm)	Total Number of Grids
Coarse	2	0.05	0.9	175,755
Medium	1	0.04	0.7	331,418
Fine	0.5	0.02	0.5	547,629

list the relevant parameters and simulation settings.

A Fourier transform is performed on the simulated lift coefficient of the iced conductor under different wind speeds to obtain the vortex shedding frequency. Figure 5 shows a comparison of the simulated vortex shedding frequencies of conductors with different ice thicknesses and the wind tunnel test results. The maximum deviation between the simulation and test results is 18% when a coarse mesh is used, the maximum deviation between the simulation and test results is 6% when a medium mesh is used, and the maximum deviation between the simulation and test results is 7% when a fine mesh is used. In general, the simulation results in this paper match well with the test results, indicating that the modeling method and the model built in this study are reliable. On the basis of a comprehensive consideration of the calculation time and accuracy, the medium mesh is used in subsequent calculations.

3. Realization of a Nonuniform Wind Field

3.1. Simulation of Fluctuating Wind Speed

The power spectrum of the fluctuating wind speed describes the distribution of the fluctuating wind energy in the frequency domain, reflecting the contribution of each frequency component in the fluctuating wind to the total kinetic energy of the turbulent fluctuations. In this work, the Davenport spectrum, which is widely used in actual engineering, is chosen [40], and its expression is as follows:

$${\displaystyle \frac{f{S}_u(f)}{u_{*}^2}}={\displaystyle \frac{4{\overline{f}}^2}{{(1+{\overline{f}}^2)}^{\frac{4}{3}}}}$$

(28)

where f is the fluctuating wind frequency, $\overline{f}=\left(fL\right)/U_{10}$; L = 1200 m; and U10 is the average wind speed at a height of 10 m.

The harmonic superposition method is used to simulate the fluctuating wind speed. By introducing a fast Fourier transform (FFT) and different interpolation techniques, the simulation time can be greatly shortened without affecting the simulation accuracy. The cross-spectral density matrix is as follows:

$$Q(\omega )=\left[\begin{array}{cccc}{Q}_{11}(\omega )& Q_{12}(\omega )& \cdots & Q_{1n}(\omega )\\ Q_{21}(\omega )& Q_{22}(\omega )& \cdots & Q_{2n}(\omega )\\ ⋮& ⋮& \ddots & ⋮\\ Q_{n1}(\omega )& Q_{n2}(\omega )& \cdots & Q_{nn}(\omega )\end{array}\right]$$

(29)

After performing Cholesky decomposition on the cross-spectral density matrix, the following is obtained:

$$Q(\omega )=H(\omega )H^{*}{(\omega )}^T$$

(30)

where H (ω) is the decomposed lower triangular matrix, and the expression is as follows:

$$H(\omega )=\left[\begin{array}{cccc}{H}_{11}(\omega )& 0& \cdots & 0\\ H_{21}(\omega )& H_{22}(\omega )& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ H_{n1}(\omega )& H_{n2}(\omega )& \cdots & H_{nn}(\omega )\end{array}\right]$$

(31)

The target process, that is, the time–history curve of the fluctuating wind speed, can be obtained by superposing these wavefunctions:

$$S_i(t)=2{\displaystyle {\sum }_{v=1}^i{\displaystyle {\sum }_{a=1}^N\left\{\right|H_{IV}({\omega }_{va})|\sqrt{\Delta \omega }\cos [{\omega }_{va}t-{\vartheta }_{va}({\omega }_{va})+{\theta }_{va}]\}}}$$

(32)

where $H_{IV}({\omega }_{va})$ is an element in the matrix, Δω is the length of the equal frequency interval, ${\vartheta }_{va}$ is a random number between the elements, and ${\theta }_{va}$ is the phase angle between different sampling points.

3.2. Implementation Process in Fluent Software

The average wind speed is set to 15 m/s, and the turbulence level is 2.3%. The harmonic superposition method is used to superpose the simulated fluctuating wind on the average wind to obtain the time–history curve of the actual wind speed, as shown in Figure 6. Considering the time-consuming FSI calculation and the agreeability with a uniform wind field, only the 30 s velocity–time curve is extracted, and transient aperiodic data are used with a sampling frequency of 50 Hz. Finally, the extracted 30 s wind speed data are written to the profile and saved in .csv format, which facilitates the input of the boundary conditions into the Fluent 2021 R1 software. The process of wind speed simulation and input is shown in Figure 7.

4. Calculation Results and Analysis

Many factors, including ice thickness, wind speed, wind attack angle, and ice shape, affect the aerodynamic characteristics of iced conductors. The model of the crescent-shaped icing noncircular cross-section conductor is shown in Figure 3b. The conductor diameter D is 30 mm, and the icing thickness H is 15 mm. Since this paper focuses on the differences in the aerodynamic characteristics of iced conductors in uniform and nonuniform wind fields when quasi-steady and unsteady theories are used, the working conditions are set as shown in

Table 3. Working conditions.

Computation Theory	Vibration Form	Wind Field	Wind Attack Angle (°)
Quasi-steady theory	No vibration of the conductor	Uniform flow Nonuniform flow	0, 30, 60, 90, 120, 150, 180
Unsteady theory	Crosswind vibration
	Along-wind vibration
	Torsional vibration

. The simulation was performed with a time step of 0.00005 s and 40 iterations per step, covering a total simulation time of 20 s. The allocation of the plan in Section 4 is shown in Figure 8 and is used to describe each type of motion analyzed.

4.1. Comparison of the Aerodynamic Characteristics of Iced Conductors in a Uniform Wind Field and a Nonuniform Wind Field When Quasi-Steady Theory Is Used

Figure 9 shows the calculation results for the drag coefficient CD, lift coefficient CL, and moment coefficient CM of iced conductors under two wind fields via quasi-steady theory. Figure 10 shows the Den Hartog and Nigol coefficients of the iced conductors under two wind fields when the quasi-steady theory is used.

As shown in Figure 9a–c, when the quasi-steady theory is used, the drag coefficient of the iced conductors under nonuniform wind field fluctuations becomes more obvious with changes in the wind attack angle. When the wind attack angles are 60° and 150°, the average drag coefficients of the iced conductors can reach 6.49 and 3.53, which are 210% and 192% higher than the corresponding values in a uniform wind field, respectively. The wind attack angle corresponding to the peak lift coefficient also increases. Figure 9d–f show that the root mean square of the aerodynamic coefficient of the iced conductor under a nonuniform wind field is greater than that under a uniform wind field; in particular, when the wind attack angle is 60°, the root mean square of the lift coefficient increases by 303% compared with that under a uniform wind field, and the root mean square of the drag coefficient increases by 8.8 times, indicating that the dispersion and variability of the aerodynamic coefficients of the conductor under a nonuniform wind field are greater.

Figure 10a shows that the Den Hartog coefficient of the iced conductor is always greater than zero in a uniform wind field, whereas in a nonuniform wind field, the Den Hartog coefficient of the iced conductor is less than zero when the wind attack angle is 5–20°, indicating that lateral galloping occurs in the conductor at this time. Figure 10b shows that the trends of the Nigol coefficient curves of the iced conductors in the two wind fields are basically the same; i.e., when the wind attack angle is 0–40° and 160–180°, torsional galloping can occur when the conductor is in the two wind fields, but the absolute torsion coefficient of the conductor in the nonuniform wind field is smaller than that in the uniform wind field.

Figure 9 also shows that the mean and root mean square values of the aerodynamic coefficients of the iced conductor are quite different when the wind attack angle is 60°. Therefore, the time–history curves of the aerodynamic coefficients of the iced conductor in the two wind fields (Figure 11) are used for analysis.

As shown in Figure 11, under a uniform wind field, the aerodynamic coefficients of the conductor reach the stable section relatively quickly, exhibiting constant amplitude and quasiperiodic characteristics. In the nonuniform flow field, owing to the influence of wind speed fluctuations, the aerodynamic coefficients of the conductor exhibit time-varying and nonstationary characteristics. The peak values of the coefficients (the maximum value of the absolute value of the amplitude) are much greater than those in the uniform flow field, and the peak drag coefficient increases significantly, i.e., the peak drag coefficient in the uniform flow field is 2.6, whereas in the nonuniform flow field, it is 6.3, which is an increase of 1.4 times. The peak torsion coefficient increases from 0.76 to 2, an increase of 1.6 times, and the peak lift coefficient increases from 2.31 to 5.38, an increase of 1.3 times. The aerodynamic coefficients of the iced conductor in the uniform flow field calculated based on traditional quasi-steady theory are relatively small, and the effects of certain factors, such as the large dispersion of the actual iced conductor aerodynamic coefficients and amplitude jumps, which can adversely affect the calculation results, are ignored. This has led to certain safety hazards in the design and vibration control of iced conductors. Therefore, the influence of wind field turbulence characteristics should be considered in the design of galloping resistance and wind-induced vibration control in actual transmission lines.

4.2. Comparative Analysis of the Aerodynamic Characteristics of Iced Conductors in a Uniform Wind Field When Unsteady Theory and Quasi-Steady Theory Are Used for Calculations

To further analyze the variation characteristics of the aerodynamic characteristics of iced conductors when unsteady theory is used, this section takes the uniform wind field as an example to calculate the aerodynamic coefficients and the Den Hartog and Nigol coefficients of iced conductors under different theories. The results are shown in Figure 12 and Figure 13.

As shown in Figure 12, the aerodynamic coefficients of the iced conductors calculated via the two theories are significantly different. Taking a wind attack angle of 60° as an example, the mean drag coefficients calculated via unsteady theory and quasi-steady theory are 7.57 and 2.18, respectively, and the mean lift coefficients are 0.3284 and 0.2073, respectively. Figure 12c shows that when the wind attack angle is 30°, the mean torsion coefficient calculated via unsteady theory is approximately 10 times greater than the value calculated via quasi-steady theory. In a uniform wind field, the dynamic aerodynamic coefficients of iced conductors calculated via unsteady theory are larger than those obtained via quasi-steady theory.

The Den Hartog and Nigol coefficients of the iced conductor in the uniform wind field calculated via the two theories are shown in Figure 13. As shown in Figure 13a, when the quasi-steady theory is used for calculation, conductor galloping does not occur; when the unsteady theory is used for calculation, the Den Hartog coefficient is less than zero when the wind attack angle is 5–15°, indicating that galloping occurs within this range. As shown in Figure 13b, when the wind attack angle is 0–25° and 160–180°, the Nigol coefficient calculated via unsteady theory is less than zero, indicating that the conductor is more prone to torsional galloping in this range; when quasi-steady theory is used, the ranges of wind attack angles at which torsion galloping of the conductor may occur are 0–40° and 160–180°, which are close to those of unsteady theory, but there are differences. For an iced conductor in a uniform wind field, the aerodynamic characteristics calculated via quasi-steady theory cannot accurately reflect the wind attack angle range at which galloping occurs, and in practice, unsteady theory should be considered when simulating the aerodynamic coefficients.

Figure 12 also shows that when the wind attack angle is 30°, the aerodynamic coefficients of the iced conductors calculated by the two theories are generally different. Therefore, the time–history curves of the aerodynamic coefficients of the iced conductor obtained via the two theories at a wind attack angle of 30° (Figure 14) are used for analysis.

As shown in Figure 14, the aerodynamic coefficients of the iced conductors in the uniform wind field calculated via the two theories quickly become stable, exhibiting constant amplitude and quasiperiodic characteristics. However, when unsteady theory is used, owing to the aeroelastic coupling effect between the conductor and the wind field, the period of the aerodynamic coefficients is larger, the fluctuation range of the amplitude is larger, and the peak values (the maximum value of the absolute value of the amplitude) of the aerodynamic coefficients are also larger than the results calculated via quasi-steady theory, with the torsion coefficient exhibiting the largest change, followed by the drag coefficient and the lift coefficient (the peak drag and lift coefficients calculated via quasi-steady theory are only 1.4 and 1.61, whereas the values are 2.3 and 2.78, respectively, via unsteady theory). Under the same physical parameters of the conductor and wind field, the aerodynamic coefficients of the iced conductor calculated via unsteady theory are all larger than the corresponding calculation values based on quasi-steady theory, indicating that the traditional method of studying the galloping problem of iced conductors via quasi-steady theory deviates greatly from the actual situation.

4.3. Comparative Analysis of the Aerodynamic Characteristics of Iced Conductors in a Nonuniform Wind Field When Unsteady Theory and Quasi-Steady Theory Are Used

In this section, the variations in the aerodynamic characteristics of iced conductors in a nonuniform wind field via the two theories are discussed, and the calculation results are shown in Figure 15, Figure 16 and Figure 17.

Figure 15 shows the aerodynamic coefficients of the iced conductors in the nonuniform wind field calculated via different theories. In a nonuniform wind field, the trends of the drag coefficient of the iced conductor versus the wind attack angle obtained by the two theories are basically the same, but those of the lift coefficient and moment coefficient are significantly different. Figure 16 shows a comparison of the time–history curves of the aerodynamic coefficients of the iced conductor in the nonuniform wind field when the wind attack angle is 30°. The aerodynamic coefficients all exhibit very high amplitude characteristics because of wind speed fluctuations. However, since unsteady theory takes the aerodynamic coupling between an iced conductor and a wind field into account, the calculated amplitudes of the aerodynamic coefficients are larger than the corresponding amplitudes based on quasi-steady theory.

Table 4. Statistical characteristics of the aerodynamic coefficients of iced conductors in a nonuniform wind field when the wind attack angle is 30°.

Different Working Conditions	Drag Coefficient			Lift Coefficient			Moment Coefficient
	Mean	SD	Peak	Mean	SD	Peak	Mean	SD	Peak
Unsteady theory	1.01	0.631	8.16	0.64	0.142	6.49	−3.92	7.75	−14.8
Quasi-steady theory	0.88	0.246	1.53	0.73	0.581	2.10	−0.44	0.181	−1.73

lists the statistical characteristics of the aerodynamic coefficients calculated via the two theories.

As shown in Table 4, the mean, standard deviation (SD), and peak values of the aerodynamic coefficients of the conductors calculated via the two theories all exhibit large differences; in particular, the statistical characteristics of the drag and moment coefficients calculated via unsteady theory are much larger than the corresponding results calculated via quasi-steady theory. For the lift coefficient, although the mean and SD calculated via unsteady theory are less than the corresponding values calculated via quasi-steady theory, the peak value can be three times the calculation result via quasi-steady theory. These results again show that the traditional method that uses only the mean values of the aerodynamic characteristics of iced conductors calculated via quasi-steady theory neglects the adverse effects of the variability in the aerodynamic coefficients of the conductor and conceals the response characteristics of the conductor in an actual wind environment, which introduces safety hazards to the reasonable design and galloping control of iced conductors. Therefore, starting from unsteady theory and simultaneously considering the effect of wind field turbulence on the aerodynamic characteristics of iced conductors is a key research direction for solving these problems.

Figure 17 shows the Den Hartog and Nigol coefficients of the iced conductors calculated via different theories in a nonuniform wind field. Figure 17a shows that in a nonuniform wind field, the difference in the Den Hartog coefficients calculated via the two theories is not significant; the range of wind attack angles at which conductor galloping occurs according to unsteady theory is slightly larger than that calculated via quasi-steady theory. In general, the Den Hartog coefficient is less than zero at a wind attack angle range of 5–20°, indicating that conductor galloping could occur in this range. Figure 17b shows that the Nigol coefficient curve calculated via unsteady theory fluctuates more severely. According to the Nigol coefficient, the wind attack angle ranges at which conductor torsional galloping occurs are as follows: 0–25° and 70–80°. According to quasi-steady theory, conductor torsional galloping occurs only at wind attack angles ranging from 0–30° to 160–180°. This shows that for an iced conductor, whether it is a uniform or nonuniform wind field, it is more reasonable to use the unsteady theory to calculate the aerodynamic characteristics and the wind attack angle range for the occurrence of conductor galloping.

4.4. Comparative Analysis of the Aerodynamic Characteristics of Iced Conductors in Uniform and Nonuniform Wind Fields When Unsteady Theory Is Used

4.4.1. Considering Only the Crosswind Vibration of the Conductor

When only the vibration of the conductor in the crosswind direction is considered, the aerodynamic coefficients of the iced conductor in a uniform wind field and a nonuniform wind field calculated via unsteady theory are shown in Figure 18, and the Den Hartog and Nigol coefficients are shown in Figure 19.

As shown in Figure 18, when unsteady theory is used and only the crosswind vibration of the iced conductor is considered, the trends of the drag coefficient and the lift coefficient of the iced conductor in the nonuniform wind field are similar, but those of the moment coefficient are quite different, and its absolute amplitude is significantly greater than that in the uniform wind field. Figure 18 also shows that the root mean square of the drag coefficient of the iced conductor in the nonuniform wind field is larger than that in the uniform wind field, whereas the changes in the root mean square values of the lift coefficient and torsion coefficient are more complicated. Figure 19a shows that in the nonuniform wind field, the Den Hartog coefficient of the iced conductor is less than zero when the wind attack angle is 5–20°, indicating that the conductor undergoes lateral galloping; moreover, the corresponding wind attack angle range in the uniform wind field is slightly smaller. Figure 19b shows that in a nonuniform wind field, the wind attack angle ranges at which torsion galloping occurs are greater, i.e., 0–25°, 70–80°, and 125–165°. In the uniform wind field, the Nigol coefficient is less than zero only when the wind attack angle is 0–25°.

Figure 20 shows the time–history curves of the aerodynamic coefficients of the iced conductor in two wind fields calculated via unsteady theory when the wind attack angle is 30°. The aerodynamic coefficients of the iced conductor in both wind fields exhibit strong nonstationary characteristics, but the two are significantly different, i.e., the time–history curves of the aerodynamic coefficients of the conductor in the uniform wind field show a more stable vibration, whereas the time–history curves of the aerodynamic coefficients of the conductors in a nonuniform wind field have obvious randomness and amplitude jumps.

Table 5. Statistical characteristics of the aerodynamic coefficients of iced conductors in different wind fields when the wind attack angle is 30°.

Different Working Conditions		Drag Coefficient			Lift Coefficient			Torsion Coefficient
		Mean	SD	Peak	Mean	SD	Peak	Mean	SD	Peak
Crosswind vibration	Uniform wind field	1.06	0.091	2.24	0.79	1.029	2.92	−2.38	3.53	−12.8
	Nonuniform wind field	1.01	0.631	8.16	0.64	0.142	6.49	−3.92	7.75	−68.2

lists the statistical characteristics of the aerodynamic coefficients of the iced conductors in the two wind fields. The mean aerodynamic coefficient of the conductor in the nonuniform wind field is close to that in the uniform wind field, but the SD and peak value increase significantly, indicating that the overall variability in the aerodynamic coefficients of the conductor in the nonuniform wind field is greater than that of a conductor in a uniform wind field; that is, the turbulent flow characteristics of the wind field have a greater impact on the aerodynamic characteristics of the iced conductor.

Figure 21 shows a comparison of the peak displacements in the crosswind direction of the iced conductors in the two wind fields when unsteady theory is used. In most wind attack angle ranges, the peak crosswind displacement of the iced conductor in the nonuniform wind field is greater; in particular, when the wind attack angle is 30°, the peak crosswind displacement of the conductor in the nonuniform wind field is close to 10D. Figure 22 shows the time–history curves of the crosswind displacement of the conductor in the two wind fields when the wind attack angle is 30°. The crosswind displacement amplitude of the iced conductor varies significantly with time in a nonuniform wind field. The overall characteristics are nonstationary and non-Gaussian, whereas the displacement amplitude of the conductor in the crosswind direction in a uniform wind field is relatively stable.

4.4.2. Considering the 3-DOF-Coupled Vibration of the Conductor in the Crosswind Direction, Along-Wind Direction, and Torsional Direction

Since an actual iced conductor may undergo crosswind vibrations in the crosswind direction, along-wind direction, and torsional direction, it is more practical to consider the coupling effect of these three vibrations. Figure 23 shows the trends of the aerodynamic coefficients of the iced conductor versus the wind attack angle under two wind fields when the 3-DOF-coupled vibration of the conductor in the crosswind direction, along the wind direction, and torsional direction is considered. The trends of the drag coefficient and lift coefficient of the iced conductors versus the wind attack angle are basically the same in the two wind fields, but those of the torsion coefficient are quite different.

Figure 24 shows the Den Hartog and Nigol coefficients of the iced conductors in the two wind fields when the 3-DOF-coupled vibration of the conductors is considered. Figure 24a shows that in a nonuniform wind field, the wind attack angle range at which lateral conductor galloping occurs is 5–20°, which is significantly larger than that in a uniform wind field. Figure 24b shows that in a nonuniform wind field, the wind attack angle range at which torsional conductor galloping occurs is 0–45°, and the corresponding wind attack angle range in a uniform wind field is 10–60°. Using unsteady theory and considering the 3-DOF-coupled vibration of conductors, the wind attack angle ranges at which iced conductor galloping occurs in the two wind fields are quite different, and these effects should be considered in the design of actual transmission lines.

Figure 25 shows the time–history curves of the aerodynamic coefficients of the iced conductors when the wind attack angles are 0°, 30°, 120°, 150°, and 180°, and the 3-DOF-coupled vibration of the conductors is considered. Figure 23 also shows that there are significant differences in the aerodynamic coefficients of the iced conductors in the two wind fields. In the uniform wind field, after the calculated aerodynamic coefficients enter the stable phase, the time–history curves also exhibit steady changes; the time–history curve of the aerodynamic coefficient of the conductor in the nonuniform wind field exhibits strong nonstationarity, and the amplitude jumps. Although the mean aerodynamic coefficients of the conductor in the two wind fields are similar, at most wind attack angles, the peak aerodynamic coefficients of the conductor in the nonuniform wind field are much greater than those in the uniform wind field, and the variability is significantly greater. Therefore, when analyzing the aerodynamic characteristics of iced conductors, in addition to the traditional mean aerodynamic coefficients, the effect of the variability in the aerodynamic coefficient on the wind-induced vibration of the conductor should also be considered.

4.4.3. Frequency Domain Analysis

To further analyze the characteristics of the aerodynamic coefficients of the iced conductor calculated via unsteady theory, this section applies the Fourier transform method to perform frequency domain analysis on these aerodynamic coefficients, and the calculation results are shown in Figure 26.

Figure 26 shows that for different wind fields, the energy distributions of the aerodynamic coefficients of the iced conductor in the frequency domain are significantly different. When the wind attack angle is 0° or 30°, the frequency range that contributes the most to the aerodynamic coefficients of the conductor in the nonuniform wind field is the low-frequency band, which is distributed in a wide frequency range; however, in a uniform wind field, the frequency range that contributes the most to the aerodynamic coefficients of the conductor is relatively small. With increasing wind attack angle, the frequency range that contributes more to the aerodynamic coefficients of the conductor in a uniform wind field is relatively concentrated. For example, when the wind attack angle is 120° or 150°, the frequencies that contribute more to the aerodynamic coefficient of the conductor are concentrated at 0.2 Hz and 0.4 Hz. In the nonuniform wind field, the frequencies that contribute more to the aerodynamic coefficients of the conductors are distributed between 0.2 and 0.3 Hz, and there is no obvious dominant frequency.

To provide a more comprehensive understanding of these dominant frequencies, we introduce the Strouhal number. The Strouhal number $S_t$ is a dimensionless quantity used in fluid dynamics to describe periodic unsteady flow. Its uniqueness lies in linking the instability of fluid motion and boundary layer separation with the relatively stable vortex shedding frequency $f_s$. The Strouhal number is defined as follows:

$$S_t={\displaystyle \frac{f_{s}L}{U}}$$

(33)

where $f_s$ is the vortex shedding frequency, L is the characteristic length, and U is the flow velocity. The following provides the Strouhal numbers under two scenarios in different wind fields, analyzing the vortex shedding frequency of the conductor, with the results shown in Figure 27.

On the basis of our calculations, the Strouhal numbers for the iced conductor under various scenarios are on the order of 10⁻⁴ or lower. This indicates that the flow is dominated by high-speed oscillations rather than vortex shedding. In such cases, viscous effects are minimal, and inertial forces prevail. The low Strouhal numbers observed suggest that the fluctuations in the aerodynamic coefficients are influenced primarily by high-frequency oscillations associated with the unsteady wind field rather than by periodic vortex shedding. This aligns with our frequency domain analysis, where no prominent peaks corresponding to vortex shedding frequencies are observed in nonuniform wind fields.

4.4.4. Analysis of the Wind-Induced Response of Iced Conductors

Figure 28 shows a comparison of displacements in the crosswind and along-wind directions and torsion angles of the iced conductors in the two wind fields. Within the wind attack angle range of 58–90°, the mean crosswind displacement of the conductor in the nonuniform wind field is less than that in the uniform wind field; in other wind attack angle ranges, the mean crosswind displacement of the conductor in the nonuniform wind field is larger than that in the corresponding one in the uniform wind field, with a maximum increase of 33%, and the trend of the crosswind displacement of the conductor with the wind attack angle is also different in the two wind fields. In the wind attack angle range of 3–50°, the mean displacement of the conductor along the wind direction in the nonuniform wind field is greater than that in the uniform wind field, with a maximum increase of 40%. In the wind attack angle ranges of 0–60° and 90–180°, the mean torsion angle of the conductor in the nonuniform wind field is much lower than that in the uniform wind field, and when the wind attack angle range is 60–98°, the mean torsion angle of the conductors in the nonuniform wind field is greater than that of the conductors in the uniform wind field.

Figure 29 and Figure 30 show the time–history curves of conductor displacements in the two wind fields when the wind attack angle is 30° and 120°, respectively, and the time–history curves of conductor displacements in the two wind fields significantly differ. In general, in a uniform wind field, the displacement amplitudes in the crosswind direction and along-wind direction of the conductor gradually stabilize within a certain range over time, with the peak displacement being only 6.5D in the along-wind direction and 8.3D in the crosswind direction. In the nonuniform wind field, the displacement response of the conductor fluctuates more violently, with large fluctuations in amplitude; the displacement in the crosswind direction reaches a maximum of 20D, the peak displacement in the along-wind direction reaches approximately 25D, and both are much greater than those in the uniform wind field. The maximum angular variation in the torsional movement of the conductor in a nonuniform wind field can reach approximately 125°, which is much greater than that in a uniform wind field. The displacement response of iced conductors in a nonuniform wind field is much greater than the response of conductors in a uniform wind field. Figure 24 shows that the galloping of iced conductors occurs more easily in a nonuniform wind field. Moreover, on the basis of the torsional displacement diagrams of the conductors in the two wind fields, in the nonuniform wind field, the torsional angle of the conductor increases greatly, which in turn increases the conductor displacement in both the along-wind direction and the crosswind direction. The coupling effect of vibration leads to torsional galloping, which in turn affects conductor displacement in both the along-wind direction and the crosswind direction.

Figure 31 shows a comparison of the displacement spectra of the iced conductors when the wind attack angles are 30° and 120°. The energy distributions of the displacement response of the iced conductor under the different wind fields notably differ in the frequency domain. In a nonuniform wind field, the energy of the conductor displacement response is distributed across a wide frequency range at low frequencies, whereas the energy distribution of the conductor displacement response in a uniform wind field is concentrated at certain frequencies. For example, when the wind attack angle is 30°, the energy of the conductor displacement response in a uniform wind field is concentrated at 6.7 Hz, whereas the energy of the conductor displacement response in a nonuniform wind field is mainly distributed in the frequency range of 0.1–0.4 Hz; there is also some displacement response energy between 6 and 8 Hz, but this energy is not as obvious as that in a uniform wind field. When the wind attack angle is 120°, the energy of the displacement response of the conductor in the uniform wind field is concentrated in the low-frequency range of 0–0.05 Hz, and the displacement in the along-wind direction and torsion angle spectral response of the conductor in the nonuniform wind field are distributed in the frequency range of 0–1 Hz, whereas the displacement spectrum response in the crosswind direction is distributed between 0 and 4 Hz. The turbulence characteristics of the wind field not only affect the response characteristics of the iced conductor in the time domain but also affect the response energy distribution in the frequency domain.

Figure 32 shows the movement trajectories of the conductor in a nonuniform wind field under different initial wind attack angles. The movement trajectories of the iced conductors vary greatly under different initial wind attack angles and have no periodicity or regularity. The maximum amplitudes in the along-wind direction and in the crosswind direction vary between 15D and 25D, indicating that galloping has already occurred at each wind attack angle.

To further elucidate the causes of the above phenomena, the velocity and pressure contours of the iced conductors in the nonuniform wind field calculated via unsteady theory are shown in Figure 33 and Figure 34. The figure shows the velocity contour, with the pressure contour as the inset.

Figure 33a and Figure 34a show the contours when the initial wind attack angle is 0°. Owing to the initially small windward area of the iced conductor, the force bearing is relatively stable; therefore, the displacement is relatively stable, and only movement in the along-wind direction occurs within a small range. At 6.5 s, the conductor starts to undergo torsional movement, so the windward area gradually increases, the force bearing gradually becomes complex, the conductor torsional movement becomes increasingly intense, and torsional movement affects the vibrations of the conductor in the crosswind direction and in the along-wind direction, resulting in significant galloping of the iced conductor.

When the initial wind attack angle is 30°, the torsional motion of the conductor is more intense, with the torsion angle approaching 100° at certain moments. At this time, the lateral vibration of the conductor is coupled to torsional motion, thus affecting the motion of the conductor and increasing the complexity of the trajectories.

When the wind attack angle is 150°, the cross-section of the conductor is twisted at 8.5 s due to wind excitation, and the angle between the conductor and the direction of incoming wind is close to 180°. Moreover, the maps after this moment show that there is no additional torsional movement, and the conductor simply maintains its status until the end of the calculation, so the trajectory after 8.5 s approaches that at a wind attack angle of 180°.

When the initial wind attack angle is 180°, there is almost no torsional movement. At this wind attack angle, the windward area of the conductor is smaller, and the shape of the conductor is more regular, so torsional movement does not easily occur; only lateral vibration occurs, the motion trajectories are periodic, and the amplitude is much lower than that at other wind attack angles.

To obtain a proper representation of the vortex structure around the trailing edge of an airfoil, it is necessary to select a suitable vortex identification method. For example, Jeong and Hussain [41] proposed the ${\lambda }_2$ criterion, which identifies the existence and characteristics of vortices by analyzing the velocity gradient of the flow field. The core idea of this method is to derive a quantity related to the velocity gradient of the flow to identify vortex structures.

In 2016, Liu et al. [42] introduced the $\Omega$ method for vortex identification. Compared with previous methods, the $\Omega$ method innovatively decomposes vorticity $\omega$ into a rotational part and a nonrotational part, thereby overcoming the issue of manual threshold tuning required in second-generation vortex identification methods. The derivation of this method is as follows:

Velocity gradient tensor decomposition:

$$\nabla V=A+B$$

(34)

where A represents the symmetric tensor and B represents the antisymmetric tensor.

Definition of the rotational proportion parameter $\Omega$:

$$\Omega ={\displaystyle \frac{\Vert B{\Vert }_F^2}{\Vert A{\Vert }_F^2+\Vert B{\Vert }_F^2}}$$

(35)

where $\Vert {\Vert }_F$ denotes the Frobenius norm, which is used to measure the strength of the tensor. To prevent division by zero, a small positive constant $ϵ$ is added to the denominator, and the corrected formula becomes:

$$\Omega ={\displaystyle \frac{\Vert B{\Vert }_F^2}{\Vert A{\Vert }_F^2+\Vert B{\Vert }_F^2+ϵ}}$$

(36)

$$ϵ=0.001\times {\left({‖B‖}_F^2-{‖A‖}_F^2\right)}_{\max }$$

(37)

The parameter $\Omega$ ranges from 0 to 1. When $\Omega >0.5$, it can be used as a criterion for vortex identification. The results are shown in Figure 35.

As shown in Figure 35a, with an initial wind attack angle of 0°, during the initial stage (t < 6.5 s), the small windward area leads to weak and symmetrically distributed vortex structures, resulting in stable displacement of the conductor, primarily along the wind direction. As time progresses (t > 6.5 s), the onset of torsional motion breaks the symmetry, causing asymmetric vortex shedding. This asymmetry amplifies torsional vibration and induces significant oscillatory motion, as alternating vortex shedding interacts with the conductor’s vibration.

When the initial wind attack angle is 150°, the vortex structures exhibit significant asymmetry from the start. During the early stage (t < 8.5 s), vortex generation and shedding occur frequently with irregular patterns, resulting in unbalanced aerodynamic forces on the conductor, which in turn lead to torsional and lateral vibrations. As time progresses, the vortex shedding process gradually stabilizes (t ≥ 8.5 s), with consistent vortex generation frequencies and patterns. At this point, the conductor becomes almost parallel to the wind direction (≈180°), leading to reduced aerodynamic force fluctuations, significantly decreased vibration amplitudes, and stabilized motion trajectories.

When the initial wind attack angle is 180°, throughout the entire simulation, the vortex structures remain small and symmetrical, reflecting the reduced windward area and the conductor’s regular shape. There is almost no significant vortex shedding, resulting in the absence of torsional vibrations. Only periodic and low-amplitude lateral vibrations occur. This indicates that when the conductor is nearly parallel to the wind direction, the aerodynamic force symmetry is maintained, reducing the generation of unbalanced forces and suppressing complex vibration patterns.

5. Conclusions

In this study, the secondary development function of Fluent software combined with moving mesh technology is used to simulate the movement of iced conductors in a wind field. The harmonic superposition method and the Davenport coherence function are used to simulate the nonuniform wind field, and a two-way FSI simulation of iced conductors under a nonuniform wind field is performed. The effects of a uniform wind field and a nonuniform wind field on the aerodynamic characteristics of iced conductors calculated via quasi-steady theory and unsteady theory are analyzed, and the main conclusions are as follows:

• According to quasi-steady theory, the trends of the mean lift coefficient and moment coefficient of the iced conductor in the uniform and nonuniform wind fields are similar, but those of the mean drag coefficient are significantly different, and the dispersion of the aerodynamic coefficients of the conductor in the nonuniform wind field is significantly greater than that in the uniform wind field.
• In a uniform wind field, the aerodynamic coefficients of an iced conductor calculated via unsteady theory are greater than those calculated via quasi-steady theory. In a nonuniform wind field, the time-varying nature, randomness, and nonstationarity of the aerodynamic coefficients of the conductors calculated via unsteady theory are very obvious. For example, the mean, SD, and peak values of the drag coefficient and torsion coefficient of conductors calculated via unsteady theory are larger than those calculated via quasi-steady theory; notably, the peak lift coefficient can approach three times that calculated via quasi-steady theory. This shows that the traditional method using quasi-steady theory to calculate the aerodynamic coefficients (the mean values are typically used) of the iced conductor conceals the real characteristics of iced conductors in an actual wind environment, making the calculated aerodynamic coefficients less than the actual values. Moreover, the adverse effects of the variability are also ignored.
• When only the crosswind vibration of the iced conductor is considered, the peak aerodynamic coefficients of the conductor in the nonuniform wind field are doubled compared with the corresponding values in the uniform wind field; the fluctuation amplitude is larger, the dispersion and variability of the aerodynamic coefficients are stronger, and the conductor displacement in the crosswind direction is larger than that in the uniform wind field.
• When the coupled vibrations of the conductor in the crosswind direction, along-wind direction, and torsional direction are considered simultaneously, the mean aerodynamic coefficients of the conductor in the nonuniform wind field are less than those in the uniform wind field, but the peak value and standard deviation (SD) are greater than those in the uniform wind field. The energy distribution of the conductor displacement response in a nonuniform wind field is within a broad frequency interval in the low-frequency range without an obvious dominant frequency. The energy distribution of the conductor displacement response in a uniform wind field has a relatively small frequency domain interval, and with increasing wind attack angle, the energy of the conductor response in a uniform wind field is concentrated within a certain frequency range.
• When the coupled vibrations of the conductor in the crosswind direction, along-wind direction, and torsional direction are considered at the same time, the mean displacements in the crosswind direction and along-wind direction and the torsion angles of the conductors in the nonuniform wind field are greater than those in the uniform wind field. Moreover, the coupling of torsional motion with motions in the crosswind direction and along-wind direction can change the galloping pattern of the conductor at certain wind attack angles, thus increasing the amplitude of conductor displacement.
• When the aeroelastic effect between the iced conductor and the airflow and the influence of wind field turbulence characteristics are considered, the conductor is more likely to experience torsion dispersion and considerable galloping. Therefore, the effects of dynamic changes in the aerodynamic characteristics and wind turbulence on the conductor should be considered when designing the wind resistance and controlling the vibration of an iced conductor.