The Effect of Continuous Trapezoidal Straight Spoiler Plates on the Vortex-Induced Vibration of Wind Turbine Towers

Abstract

This paper proposes a method of controlling the vortex-induced vibration (VIV) of wind turbine towers by adding continuous trapezoidal straight spoiler plates (TS) onto their outer surface: a fluid–solid coupling model was constructed to simulate the processes of Karman vortex generation and shedding on the different surfaces of an original tower (O–tower) and a tower with TS (TS–tower) with assumed and actual Re, while the VIV frequencies were also calculated and compared; the effects of the TS geometry parameters on the VIV frequency of towers were studied to investigate the recommended size; a modal analysis was carried out to research the effects of TS on the vortex-induced resonance risk of towers; and the simulation results as well as relevant research conclusions were validated by an analogical wind tunnel test.

1. Introduction

A wind turbine tower is the vital support component connecting the nacelle and platform. The main function of a tower is supporting the rotor system and absorbing the vibration derived from the wind load. With the rapid development of wind turbines and the increasing demand for electricity, the towers are becoming taller, but, because of cost limitations, they are also becoming thinner, which makes the influence of wind-induced vibration more and more obvious [1]. Kang L [2] found that large structures with a slenderness ratio greater than five or an eigenfrequency less than 0.2 Hz are prone to large-scale vibration. P. Mendis [3] et al., proposed that the lateral vibration of high tower occurs easily, which is caused by VIV. Wind turbine towers are usually tall, cylindrical structures with a smooth surface, and their length-to-diameter ratio is generally 20–50, which is destined to be easily affected by VIV.

Figure 1 indicates the mechanism of VIV generation by towers. If the wind flows through the surface of a tower at particular regimes (i.e., lower speeds, laminar), the airflow will generate periodic vortices on both sides of the tower; consequently, the fluid pressure will be varied when the vortices are shedding from a tower’s surface, and the extra load excitation derived from such differential pressure causes a tower to vibrate periodically. Ulrik [4] found that when the excitation frequency of wind is close to or even coincides with the natural frequency of a wind turbine tower, the tower will produce huge vibration and even collapse directly. A. Chizfahm et al. [5] applied Euler–Bernoulli beam theory and the Galerkin method to establish a nonlinear parameter model of a bladeless wind turbine, and applied this model to study the influence of wind speed on the dynamic characteristics of a wind turbine. It was found that, if the frequency of vortex shedding was synchronized with the eigenfrequency of a wind turbine tower, the vibration amplitude would increase significantly, which indicated “vortex-induced resonance” harmful to the reliability of a wind turbine. N. Khodaie et al. [6,7,8,9] proposed that structures with high flexibility, low damping, and a large aspect ratio are easily affected by VIV, resulting in the fatigue damage of structures. Chou [10] et al. analyzed the failure mode of wind turbine towers and concluded that VIV is the main cause of tower structural damage. The failure of the Ferrybridge power plant in 1960 in Yorkshire, England and the Tacoma Bridge in 1940 in Washington, WA, the United States are examples of resonance failure due to vortex-induced vibration that cannot be ignored [11]. Considering the great influence of VIV on the structural safety of wind turbine towers, it is seriously significant to investigate effective solutions for reducing VIV. In reality, wind turbine towers very much resemble chimneys and industrial stacks, and some of the literature and international standards have provided certain methodologies for resolving VIV issues of similar structures. Generally speaking, there are three common methods to reduce the influence of VIV on large structures:

Structural modification: The modification of the size of a large structure can change the eigenfrequency and keep it away from the possible VIV frequency. However, due to the large size of wind turbine towers, the method of structural modification is very difficult and will increase construction costs. Therefore, great attention has been paid in engineering to the two methods of auxiliary dampers and aerodynamic shape modification [12].

Active and passive dampers: Active dampers regulate the damping and stiffness of structures through hydraulic and electromechanical systems, thereby suppressing the VIV of structures. One of the most representative of these is an active mass damper (AMD); the essence of passive dampers is mass–spring–damper systems, which can reduce vibration by changing the resonance characteristics of structures, among which a tuned mass damper (TMD) is the most widely used. Researchers have found that the reasonable configuration of an AMD [13,14,15] and a TMD [16,17,18] can reduce the vortex-induced response of the structure and reduce the occurrence of instability. However, the design of auxiliary dampers must match the mass and stiffness of structures themselves, otherwise it will aggravate the vibration of structures, the auxiliary dampers will not weaken the existing vortex-induced load, and the fatigue damage problem of structures will not be effectively solved.

Aerodynamic shape modification: At present, the most-studied method of reducing vibration is aerodynamic shape modification. The common principle is to change the flow field around structures by adding vortex-disturbing devices on the surface of structures; these devices can destroy the formation and shedding of a vortex in a wake to increase the stability of fluid flow. Figure 2 shows several vortex-disturbing devices, such as strakes, spoilers and TS, etc. [19,20,21]. Vegard Holland et al. [22] used CFD to study the hydrodynamic performance of strakes on full-scale offshore platforms. The calculation results show that the strakes reduce the vorticity behind the platform and the transverse force, but the disadvantage is that the resistance is increased. Gustavo R.S. Assi et al. [23] studied the effect of the unconventional geometries of strakes on the vortex-induced vibration of low-mass and structural-damping cylinders. It was found that traditional straps can reduce the vibration amplitude by 88% and the drag force by 48%, while zigzag straps can reduce the vibration amplitude by 95% and the drag force by 54%. The separation mechanism of a wake is explained by visualizing the flow field. Sukarnoor et al. [24] studied the vortex-induced vibration of rigid cylinders in a series at critical spacing. The results show that spiral plates can reduce the vibration amplitude of upstream and downstream cylinders, but that the arrangement of plates affects the suppression effect of vortex-induced vibration. Zhou et al. [25] studied the vortex-induced vibration of smooth cylinders, cylinders with grooves, or concave holes. It was found that the average drag and lift of rough cylinders were greatly reduced compared with those of smooth cylinders, and it was considered that surface roughness reduced the intensity of vortex-induced vibration. Bianchi Valerio et al. [26] used splitter plates to carry out parameterized experimental research on the suppression of the VIV of a low-aspect-ratio cylinder. The research shows that the splitter plate structure can significantly reduce the vibration amplitude in the flow direction and transverse direction of a cylinder. Goncalves [27] also proved this point through experiments. Lubbad [28] et al. studied the efficiency of strakes to suppress cylinder vibration, and found that different strake sizes can reduce the flow direction and transverse amplitude of a cylinder. Quen et al. [29] studied the suppression effect of the pitch and height of strakes on the VIV of a cylinder. The research found that an increase in pitch will delay the locking, and that the height of strakes has a greater impact on the amplitude of the VIV. Gao [30] also came to the same conclusion. Feng Xu [31] studied the problem of suppressing the vortex-induced vibration of an elastic cylinder with a travelling wave wall in a two-dimensional flow state by using a fluent, and analyzed the variation laws of cylinder displacement, the center-of-mass trajectory, and the lift and drag on the surface of a cylinder in detail. The results show that small-scale vortices will form in the groove at the tail of a cylinder, which can control the separation of the flow field from the cylinder and eliminate the wake oscillation, thus suppressing the vortex-induced vibration. No matter whether in uniform flow or shear flow, the height of strakes has an obvious influence on the vibration amplitude, thus affecting the life of a structure. However, due to the discontinuous structure of spoiler plates and the low inclination angle of strakes, the working efficiency is generally low. Therefore, according to the theory of flow around circular cylinders, this paper investigated the mechanism of VIV and the influence on the reliability of towers by using CFD, and proposed a method of adding TS onto the outer surface of towers to suppress the VIV frequency. The research conclusions should be significant for wind turbine towers to avoid vortex-induced resonance.

Figure 3 illustrates the structure and dimension definitions of the TS of towers. The significant dimensions of the TS include plate thickness (L1), height (L2), width (L3), spacing (L4), which is equivalent to the number of TS, (n), angle (θ), and the fillet (α).

2. Theoretical Basis

2.1. Flow Control Equation and Fluid Parameters

To consider three-dimensional incomprehensible fluids, the rule of fluid flow can be described by continuity Equation (1) and Navier–Stokes Equation (2); the expression of the equation in the Cartesian coordinate system is as follows:

$$\frac{\partial {\mathsf{U}}_{\mathsf{x}}}{\partial \mathsf{x}}+\frac{\partial {\mathsf{U}}_{\mathsf{y}}}{\partial \mathsf{y}}+\frac{\partial {\mathsf{U}}_{\mathsf{z}}}{\partial \mathsf{z}}=0$$

(1)

$$\frac{\partial {\mathsf{U}}_{\mathsf{x}}}{\partial \mathsf{t}}+{\mathsf{U}}_{\mathsf{x}}\frac{\partial {\mathsf{U}}_{\mathsf{x}}}{\partial {\mathsf{U}}_{\mathsf{x}}}+{\mathsf{U}}_{\mathrm{y}}\frac{\partial {\mathsf{U}}_{\mathsf{x}}}{\partial {\mathsf{U}}_{\mathsf{y}}}+{\mathsf{U}}_{\mathsf{z}}\frac{\partial {\mathsf{U}}_{\mathsf{x}}}{\partial {\mathsf{U}}_{\mathsf{z}}}=-\frac{1}{\rho }\frac{\partial p}{\partial \mathsf{x}}+\mathrm{v}(\frac{{\partial }^2{\mathsf{U}}_{\mathsf{x}}}{\partial {\mathsf{x}}^2}+\frac{{\partial }^2{\mathsf{U}}_{\mathsf{x}}}{\partial {\mathsf{y}}^2}+\frac{{\partial }^2{\mathsf{U}}_{\mathsf{x}}}{\partial {\mathsf{z}}^2})$$

(2)

$$\frac{\partial {\mathsf{U}}_{\mathsf{y}}}{\partial \mathsf{t}}+{\mathsf{U}}_{\mathsf{x}}\frac{\partial {\mathsf{U}}_{\mathsf{y}}}{\partial {\mathsf{U}}_{\mathsf{x}}}+{\mathsf{U}}_{\mathsf{y}}\frac{\partial {\mathsf{U}}_{\mathsf{y}}}{\partial {\mathsf{U}}_{\mathsf{y}}}+{\mathsf{U}}_{\mathsf{z}}\frac{\partial {\mathsf{U}}_{\mathsf{y}}}{\partial {\mathsf{U}}_{\mathsf{z}}}=-\frac{1}{\rho }\frac{\partial p}{\partial \mathsf{y}}+v(\frac{{\partial }^2{\mathsf{U}}_{\mathsf{y}}}{\partial {\mathsf{x}}^2}+\frac{{\partial }^2{\mathsf{U}}_{\mathsf{y}}}{\partial {\mathsf{y}}^2}+\frac{{\partial }^2{\mathsf{U}}_{\mathsf{y}}}{\partial {\mathsf{z}}^2})$$

(3)

$$\frac{\partial {\mathsf{U}}_{\mathsf{z}}}{\partial \mathsf{t}}+{\mathsf{U}}_{\mathsf{x}}\frac{\partial {\mathsf{U}}_{\mathsf{z}}}{\partial {\mathsf{U}}_{\mathsf{x}}}+{\mathsf{U}}_{\mathsf{y}}\frac{\partial {\mathsf{U}}_{\mathsf{z}}}{\partial {\mathsf{U}}_{\mathsf{y}}}+{\mathsf{U}}_{\mathsf{z}}\frac{\partial {\mathsf{U}}_{\mathsf{z}}}{\partial {\mathsf{U}}_{\mathsf{z}}}=-\frac{1}{\rho }\frac{\partial p}{\partial \mathsf{z}}+v(\frac{{\partial }^2{\mathsf{U}}_{\mathsf{z}}}{\partial {\mathsf{x}}^2}+\frac{{\partial }^2{\mathsf{U}}_{\mathsf{z}}}{\partial {\mathsf{y}}^2}+\frac{{\partial }^2{\mathsf{U}}_{\mathsf{z}}}{\partial {\mathsf{z}}^2})$$

(4)

In the equations, U_x, U_y, and U_z are the fluid velocity components in the three horizontal directions, t is the flow time of the fluid, ρ is the density of the fluid, p is the pressure of the fluid, and v is the kinematic viscosity of the fluid.

The Reynolds number (Re) and the Strouhal number (St) are important dimensionless parameters in the flow around circular cylinders. The Re reflects the flow state of the fluid, and the St is the similarity criterion utilized to characterize the unsteady flow. The Re and St determine the VIV frequency of the structure together:

$$\mathsf{R}\mathsf{e}=\frac{\mathsf{\rho }\mathsf{U}\mathsf{D}}{\mathsf{\mu }}$$

(5)

$$\mathsf{S}\mathsf{t}=\frac{\mathsf{f}\mathsf{\times }\mathsf{D}}{\mathsf{U}}$$

(6)

where U is the flow velocity of the fluid, D is the diameter of the cylinder, μ is the dynamic viscosity of the fluid, and f is the VIV frequency.

The aerodynamic forces acting on the tower can be divided into the lift force (F_l) and drag force (F_d); the direction of the lift is perpendicular to the flow direction of the fluid, and the direction of the drag is consistent with the flow direction of the fluid. The lift and drag are as periodic as the vortex shedding in the flow around a cylinder, the changing frequency of F_l and F_d is the same as the frequency of vortex shedding, and the phases are identical. According to the theory of flow around circular cylinders, the ratios of the lift force and drag force per characteristic length (the diameter of a tower, D) to the dynamic pressure of airflow are defined as the lift coefficient (C_l) and drag coefficient (C_d). These two important coefficients can be obtained as shown in Equations (7) and (8):

$${\mathsf{C}}_{\mathsf{l}}=\frac{\mathsf{2}{\mathsf{F}}_{\mathsf{1}}}{\mathsf{\rho }{\mathsf{U}}^{\mathsf{2}}\mathsf{D}}$$

(7)

$${\mathsf{C}}_{\mathsf{d}}=\frac{\mathsf{2}{\mathsf{F}}_{\mathsf{d}}}{\mathsf{\rho }{\mathsf{U}}^{\mathsf{2}}\mathsf{D}}$$

(8)

2.2. Control Equations of Cylindrical Vibration

The vibration of structures, induced by fluid flow, can be regarded as a mass–spring–damper system, so the control equations of a forced vibration system can be expressed as:

$$\left[\mathsf{M}\right]\left(\ddot{\mathsf{x}})+\left[\mathsf{C}\right]\right(\dot{\mathsf{x}})+\left[\mathsf{K}\right](\mathsf{x})=(\mathrm{F})$$

(9)

where [M], [C], and [K] are the mass matrix, damping matrix, and stiffness of components; (F) is the external excitation matrix; and _{$\left(\mathsf{x}\right)$} is the displacement vector, and can be expressed as a simple harmonic function:

$$\left(\mathsf{x}\right)=\left(\mathsf{A}\right)\sin \left(\mathsf{\omega }\mathrm{t}+\mathsf{\phi }\right)$$

(10)

When the component is vibrating without any extra exciting load, the external excitation matrix (F) should be 0. In this situation and according to Equation (10), the control equations can be written as follows:

$$\left(\mathsf{A}\right)\left(\left[\mathsf{K}\right]-{\mathsf{\omega }}^2\left[\mathsf{M}\right]+\mathsf{\omega }\left[\mathsf{C}\right]\right)=0$$

(11)

where $\left(\mathsf{A}\right)$ is the vibration amplitude matrix of the components and must be nonzero. This implies that Equation (11) can be established, as long as:

$$\left(\left[\mathsf{K}\right]\mathsf{-}{\mathsf{\omega }}^2\left[\mathsf{M}\right]+\mathsf{\omega }\left[\mathsf{C}\right]\right)=0$$

(12)

To solve Equation (12), the eigensolutions ${\mathsf{\omega }}_1,{\mathsf{\omega }}_2\cdots {\mathsf{\omega }}_n$ are eigenfrequencies of each mode of the system; the mode shapes can also be obtained by substituting the eigensolutions into Equation (11). In general, eigenfrequencies and mode shapes are inherent properties of mechanical structures; they are only related to mass distribution and constraint type, so any external load (such as the wind load on blades and VIV excitation) does not make sense to the modal characteristics of wind turbine systems.

3. Computational Model and Verification

3.1. Computational Fluid Domain

The main purpose of the simulation in this paper is to study the influence of the TS on the VIV frequency of a wind turbine tower, the actual diameter (D) and height (H) of the tower in this case being 4 m and 80 m, respectively. To simplify the simulation model and decrease the workload of the solving process, the simulation was carried out on the basis of a partial section model of the full-scale tower. Figure 4 describes the detailed definitions of the computational fluid domain for the simulation of VIV, and the specific data are listed in

Table 1. Boundary conditions of the computational fluid domain.

Diameter of tower (D)	4 m	Inlet velocity	10 m/s
Height of fluid domain	1.25 D	Outlet static pressure	0 MPa
Length of fluid domain	20 D	Solving time	100 s
Wide of inlet/outlet	7.5 D	Time step	0.1 s

.

To obtain the VIV frequency of the tower, which is derived from a vortex, a pressure-monitoring point is defined on a position close to the tower (as shown in Figure 4); the pressure variation of this point is monitored to obtain the frequency of vortex generation and shedding through FFT, so vortex generation is the most important condition for investigating VIV. In reality, it is difficult to simulate the processes of vortex generation and shedding from a tower surface accurately under the fluid state with a large Re (or turbulence), but the processes must be more obvious in laminar flow conditions. Consequently, the Re is set to 100 for assumption to clarify the vortex generation and shedding processes in the simulation, and the influence of TS on the VIV frequency can be investigated definitely. Furthermore, some specific simulation settings of the CFD model are considered as follows:

• Computing algorithm: laminar;
• Transient scheme: second-order backward Euler;
• Convergence criteria: the residual type is RMS, and the residual target is 0.001;
• To simulate the fluid state around the towers, the partial fluid domain which contains the monitor point is meshed as a boundary layer;
• The computational fluid domain is initialized with velocity–pressure conditions.

According to the common theory of flow around circular cylinders, the fluid will produce a vortex when it comes into contact with the tower surface, and the variations in fluid velocity and pressure around the tower can indicate the generation of a vortex, so to simulate such variation clearly is the main target of the computational model. Furthermore, the state of the vortex must be relative to the TS because the roughness of the tower surface is varied; the influence can be discussed according to the fluid streamline result. Figure 5 indicates that the turbulence around the TS is not very obvious under laminar flow conditions, so the case study does not consider the effects of turbulence on vortex generation and the VIV frequency.

3.2. Meshing

To simulate vortex generation and shedding processes as accurately as possible, the model must be meshed by high-quality elements. The elements of the fluid domain around the tower should be high-density in order to capture the pressure variation of the pressure-monitoring point effectively; the elements of other fluid domains can be sparse gradually to save simulation time. Figure 6 shows the elements of the meshed computational fluid domain, where (a) is the high-density elements around the O–tower wall with a smooth outer surface, and (b) is the elements around the TS–tower wall with spoiler pates. The preprocessing is carried out by using Hypermesh software, and the total number of elements totals almost 920,000. The fluid elements are meshed under the following quality control parameters:

• All elements are first-order tetrahedral, and the type is Solid185;
• Aspect ratio > 5;
• Tetrahedral collapse ratio < 0.5;
• Equiangle skew ratio > 0.7;
• Volume skew ratio > 0.95;
• Jacobian < 0.7.

After the element quality check process, the unqualified rates of all of the control parameters must be controlled within 1%. In this instance, the simulation results after convergence will not be influenced significantly by variation in element density; consequently, the quality of the fluid elements can be considered to satisfy accuracy requirements.

3.3. Plausibility Verification

It is necessary to verify the accuracy of the proposed methodology by calculating the typical flow state parameters of the simulated fluid field on the basis of the O–tower model. Figure 7a illustrates the time–history curves of the C_l and C_d; the results indicate that the value of the C_l is within the range of −0.319 to 0.319, the average value of C_d is 1.36, and that they become regular and periodic within a period of 2.5 s. The reason for these phenomena should be the periodicity of vortex generation and shedding.

Table 2. Reliability verification of the simulation.

Researcher	C1	Cd	fw
Braza [33]	±0.4	1.37	0.418
Mittal [34]	±0.319	1.37	0.418
This paper	±0.319	1.36	0.399

lists the simulation results of the C_l and C_d in this research in addition to theoretical results under completely consistent conditions from other examples in the literature. The numerical comparisons indicate that the obtained simulation results in this paper are basically accordant with the theoretical research results. Moreover, Professor Roshko [32] has proposed the following regular equation for determining the St when the Re is within the range of 40 to 150 by using a data-fitting method:

$$\mathsf{S}\mathsf{t}=0.212(1-\frac{21.2}{\mathsf{R}\mathsf{e}})$$

(13)

According to Equation (13), the theoretical value of the St in this case should be 0.167 (assumed Re = 100). As is shown in Figure 6b, the simulated value of f_w (0.399 Hz) can be obtained by fast Fourier transform (FFT), and the theoretical value of f_w should be 0.418 according to Equations (6) and (13). Consequently, the mentioned calculations indicate that the simulation results in this research are very similar to the theoretical results, the reliabilities of the model and methodology can be verified, and that the following simulations that focus on TS models should be accurate enough.

4. Numerical Simulation of Vortex-Induced Vibration of the TS–tower

According to the formation mechanism of VIV, it can be inferred that the VIV frequency should be directly related to the roughness of the tower’s surface. Therefore, TS can be installed on the outer surface of the tower to disturb the vortex generation, so as to reduce the VIV frequency. The relevant literature review indicates that there are very few works in the literature that focus on the influence of TS on the VIV of wind turbine towers; therefore, in this section, an investigation regarding this topic is carried out. The geometric parameters of the tower and TS are listed in

Table 3. Geometry parameters of the tower and TS.

D (m)	H (m)	L2 (mm)	L3 (mm)	θ (degree)	n (piece)	α (mm)
4	5	50	150	40–85	0–50	6

, and the θ and n (L4) parameters are varied to analyze the influence of the TS size on the VIV frequency of the tower, so as to conclude the integrated influence of the TS on the VIV characteristics of the wind turbine tower.

4.1. Influence of the θ Parameter on the VIV Frequency

Figure 8 shows the variation in the VIV frequency of the TS–tower when the θ parameter varies within the range between 40° and 85°.

The following conclusions are drawn, according to Figure 8:

1.

The VIV frequency of the O–tower f_w is 0.399 Hz, and the VIV frequencies of TS–towers with different θ parameters (f) are much lower than that of the O–tower;

2.

When θ varies within the range of 40° to 53°, the value of f is steady;

3.

When θ is larger than 53°, the value of f varies in an obvious manner, and the minimum f is 0.262 Hz when θ is 61°; the VIV frequency is 34.3% lower than that of the O–tower;

4.

When θ changes within the range of 61° to 64°, the value of f increases sharply, and the maximum f is 0.367 Hz when θ is 63°; it is also less than f_w;

5.

When θ changes within the range of 64° to 85°, the value of f has limited variation between 0.322 Hz and 0.342 Hz.

On the basis of the above discussions, it can be concluded that the spoiler plates are effective in reducing the VIV frequency of towers; the θ parameter should be designed as less than 63° to obtain optimized effectiveness; and a θ parameter that will correspond to the minimum VIV frequency must exist.

Figure 9 illustrates the vorticity simulation of the O–tower (a) and the TS–tower (b–i) with a variable θ parameter. It indicates that TS on the surface of towers can influence the situation of vorticity in an obvious manner, the number of vortices generated behind the tower are decreased significantly, and that the VIV frequency must also be decreased correspondingly.

4.2. Influence of the n Parameter on the VIV Frequency

Figure 10 shows the variation in the VIV frequency of the TS–tower when the n parameters vary within the range of 0 to 50 pieces.

Figure 10 illustrates that f_w is 0.399 Hz when n is 0 (O–tower), and that the VIV frequencies, f, are also weakened in an obvious manner by adding TS onto the tower’s surface. According to the increase in the n parameter of the circumference of the tower from 0 to 50, the varying trend in the VIV frequency, f, displays a monotone decreasing function; the result figure also indicates that, to obtain an optimized VIV frequency with decreased effectiveness, the n parameter should be designed as more than eight.

Figure 11 illustrates a vorticity simulation of the O–tower (a) and TS–tower (b–e) with variable n parameters. It indicates that the speed of the vortex generation and shedding of the TS–tower decreases significantly. The reason for this is that the roughness of the tower’s surface is increased by adding TS, which can disturb the flow state of the fluid on the tower’s surface. Therefore, the VIV frequency of the TS–tower should be far less than that of the O–tower.

4.3. Further Comparison and Discussion

Figure 12 illustrates the time–history curves of the C_l and C_d of the TS–tower. The results indicate that the value of the C_l is within the range of −0.05 to 0.05, the average value of the C_d is 0.25, and that the general trend in the time–history is similar to that of the O–tower. The reduction rates of the C_l and C_d after the addition of TS have achieved values of 84.3% and 81.6%, respectively. According to the research conclusions of Gustavo R.S. Assi et al. [23], the S45-serrated 45-strake model can supply a 55% lift reduction and 57% drag reduction; comparison of the reduction rates validates and evaluates the effectiveness of reducing VIV due to the addition of TS.

The fundamental point of the proposed methodology is to simulate the processes of vortex generation and shedding, so an assumed Re = 100 was defined to clarify the vortex characteristics under laminar flow conditions. It is still essential to carry out a simulation with actual Re to consider real flow state and to analyze the VIV problem under turbulence flow conditions, and to discuss the relationships between the effect of the addition of TS on reducing the VIV frequency and Re variation. For further investigation, the algorithm of realizable k-epsilon should be used; the actual Re can be calculated according to Equation (5), and is defined as 360,000 in this case. Figure 13 shows the variation in the VIV frequencies of the O–tower and TS–tower when the θ parameter varies within the range of 45° to 80°:

Figure 13 indicates that the effects of the TS angle on reducing the VIV frequency is still obvious under turbulence flow conditions, but there are some difference between the result figures: the VIV frequency of the O–tower (0.8 Hz) increases by 100% and the maximum VIV frequency of the TS–tower (0.599 Hz, when θ = 65°) increases by 60%; the varying trend in the TS–tower line is obviously different from the one in Figure 8; the corresponding value of the TS angle of minimum frequency (0.5 Hz) changes to 55°; and the frequency value starts to increase in an obvious manner up to the maximum value from this angle. The mentioned difference indicates that the VIV frequency of towers is relative to the Re directly, but the frequency-reduction effect can also be performed by the addition of TS. To consider the influence of the TS angle on reducing the VIV frequency under different fluid state conditions, the recommended TS angle should be less than 60°; the parameter scheme could ensure that the VIV frequency reduction effect is significant enough for wind turbine towers under practical fluid state conditions.

4.4. Modal Analysis of Wind Turbine Towers

The phenomenon of vortex-induced resonance will cause serious damage to wind turbine towers: it will occur when the VIV frequency is consistent with or close to the eigenfrequency of a wind turbine, so it is significant to carry out a modal analysis for wind turbine towers to calculate the eigenfrequencies and mode shapes. In this paper, the main purpose of the modal analysis was to investigate the effects of the addition of TS on the eigenfrequencies and mode shapes of towers relatively; it is not very essential to consider some common conditions of the two towers specifically, such as the structure of nacelles and blades, some local stiffeners (i.e., inside flanges), the gyroscopic effect of rotating blades, etc., because the influences of such conditions on the modes of the two towers are identical, except for the TS. Consequently, a simplified model of a wind turbine is established for the modal analysis, as displayed in Figure 14. The fixed support is defined on the bottom surface of a tower without considering the influence of other external loads [35], the rigid connection element is used to simulate the connection relationship between the nacelle and the tower, and a lumped mass element, Mass21, is used to describe the mass of the nacelle and blades–hub system; the assumed value is defined as 455,000 kg according to the common weight data of such a megawatt wind turbine. The tower is the key object of the modal analysis; its mass and flexibility should be determined by the structure and material, so the material parameters, such as the elastic module and Poisson’s ratio, are defined as steel (E = 206 GPa and ε = 0.3).

Table 4. The geometry parameters of wind turbines with the O–tower and TS–tower.

	Height (m)	Diameter (m)	L1 (mm)	L2 (mm)	L3 (mm)	θ (Degree)	n (Piece)	Weight (t)	Mass21 (t)
O–tower	80	4	40	—	—	—	—	31.5	455
TS–tower	80	4	16.5	50	150	61	50	31.5	455

lists the geometry parameters of wind turbines with the O–tower and TS–tower.

For the analysis, the modal results of a maximum of 30 modes were calculated, and Figure 15 and Figure 16 illustrate the first five mode shapes of the O–tower and TS–tower. According to the vibration theory of wind turbines, the higher module response attenuates rapidly due to the effect of structural damping, so the influence of the higher module is not considered in this paper.

Table 5. Comparison of the eigenfrequency and VIV frequency of towers.

Mode	O–tower		TS–tower
	Eigenfrequency (Hz)	Mode Shapes	Eigenfrequency (Hz)	Mode Shapes
1	0.4462	Radial oscillating	0.4535	Radial oscillating
2	3.9831	Radial bending	2.8794	Harmonic
3	6.7804	Harmonic	3.9641	Radial bending
4	7.4811	Radial oscillating	4.3289	Harmonic
5	9.3686	Harmonic	7.0978	Harmonic
VIV Frequency	0.399 Hz		0.262 Hz

lists the comparison of the first five eigenfrequencies and VIV frequencies of the two types of wind turbine towers; the data indicate that the eigenfrequencies are not significantly different. According to the above discussions, the VIV frequency of the TS–tower should be less than that of the O–tower, meaning that the exciting frequency of the TS–tower is further away from the eigenfrequency of the wind turbine than that of the O–tower, which greatly decreases the possibility of vortex-induced resonance.

4.5. Wind Tunnel Experimental Study

A wind tunnel experimental study was carried out to validate the effectiveness of the TS–tower in reducing the VIV frequency. According to the mechanism of vortex-induced vibration, the generated vortices will alternately fall off periodically on both sides of the tower when fluid flows through the tower surface; the process causes the fluid pressure on both sides of the tower to vary alternately, with a phase difference of half a cycle. The frequency of the pressure variation is the VIV frequency of the tower, and the data can be collected by a differential pressure sensor [36]. Figure 17 displays the structure and components of the wind tunnel test rig: the air compressor provides uniform and stable air flow to the steady-flow tube and measurement tube with a velocity of 10 m/s; the measurement model (small-scale O-specimen and TS-specimen models, as shown in Figure 18) and differential pressure sensor are installed in the experimental tube; and the geometry parameters of the measurement models are listed in

Table 6. Geometry parameters of the measurement models.

Model	Height (mm)	Diameter (mm)	L2 (mm)	L3 (mm)	n (Piece)	θ (Degree)
O-spec.	100	80	—	—	—	—
TS-spec.	100	80	1	3	50	61

. For comparison with the actual sizes of the O–tower and TS–tower in Table 4, the diameter and height of the test specimens used in the experiment are reduced by 50 times (D = 80 mm and H = 100 mm). On the basis of the model similarity criterion, the TS sizes L2 and L3 are also reduced by 50 times correspondingly. At the same time, the TS numbers and angle remained as 50 pieces and 61 degrees, respectively, because the two parameters are not related to the size difference.

Figure 19 displays the voltage signals of the differential pressure sensor with a sampling time of 10 s, which means that the number of sampling points was 20,480. The voltage signal variation tendency reflects the variation in the pressure difference on both sides of the specimens; the VIV frequency spectrums of the measurement models can be obtained by FFT, as shown in Figure 20.

According to the FFT spectrum results, the VIV frequencies of the O-specimen (f_sw) and TS-specimen (f_s) are 5.562 Hz and 3.345 Hz (the frequencies that correspond to the maximum power spectral density of air pressure signals), respectively, and a reduction rate of 40.53% is reached, similar to the reduction rate obtained by the simulation analysis (34.3%). The experimental results of the wind tunnel testing validate that the TS can reduce the VIV frequency of the tower effectively. In summary, the assumption that the TS structure could be used to reduce the VIV frequency of the tower has been analyzed from three avenues: theoretical analysis, numerical simulation analysis, and experimental verification. All of the results indicate that it is reliable to decrease the VIV exciting frequency by manufacturing TS to disturb the generation and shedding of the vortex, that the VIV frequency can be determined by a differential pressure sensor, and that the proposed methodology must be effective to avoid the VIV resonance of wind turbine towers.

5. Conclusions

This paper proposed a method to simulate the vortex-induced vibration (VIV) of wind turbine towers according to the theory of cylinder flow, and investigated the suppressive effect of TS on the VIV frequency of towers on the basis of a fluid–solid coupling analysis. The research conclusions mainly include the following four points:

1.

The proposed model can be used to analyze the vortex-induced vibration (VIV) problems of wind turbine towers. The reliability of the algorithm is validated by consulting relevant theoretical results and data from the literature (see Table 2).

2.

Manufacturing continuous trapezoidal straight spoiler plates (TS) on the outer surface of wind turbine towers can decrease the VIV frequency significantly. To obtain the most optimized effectiveness, the recommended geometry parameters of the TS are: an angle, θ, of less than 60°; a number of TS, n, greater than 8; L3 and L4 being relative to the tower diameter, D, and n; as well as L1 and L2 being relative to the thickness of the base plate (see Figure 2).

3.

The TS do not influence the eigenfrequencies of wind turbine structures significantly, but can decrease the exciting frequency (VIV frequency) in an obvious manner. It is effective for avoiding the vortex-induced resonance risk of wind turbine towers.

4.

The function of reducing the VIV frequency of TS is validated by an analogical wind tunnel test that uses small-scale specimens (with the consideration of the geometry similarity). The results indicate that the experimental VIV frequency deduction rate reached 40.53%, similar to the rate of this simulation (34.3%).

In future work, the methodology of research will be optimized to investigate the VIV situation in the fluid state with a practical Re, and will try to simulate vortex generation and shedding in turbulence more accurately. The influences of other geometry parameters of TS, such as the height, L2; width, L3; and fillet, α, on the VIV frequency will also be studied, and there will be an attempt to establish a database regarding the rule of the influence of each TS dimension variable on the VIV frequency of towers, so as to optimize the size of spoilers and minimize the influence of VIV on towers. The research conclusions obtained are significant to improve the operation stability and power generation efficiency of wind turbines; they also can be used as a reference to solve the VIV problems of other structures with similar shapes, such as offshore risers, offshore platforms, and bridges.