Effect of Rotor Thrust on the Average Tower Drag of Downwind Turbines

Abstract

A new analysis method to calculate the rotor-induced average tower drag of downwind turbines in the blade element momentum (BEM) method was developed in this study. The method involves two parts: calculation of the wind speed distribution using computational fluid dynamics, with the rotor modeled as a uniform loaded actuator disc, and calculation of the tower drag via the strip theory. The latter calculation considers two parameters, that is, the decrease in wind speed and the pressure gradient caused by the rotor thrust. The present method was validated by a wind tunnel test. Unlike the former BEM, which assumes the tower drag to be constant, the results obtained by the proposed method demonstrate much better agreement with the results of the wind tunnel test, with an accuracy of 30%.

1. Introduction

Horizontal axis wind turbines are categorized as upwind and downwind turbines according to the position of the rotor relative to the tower. Among these, upwind turbines have been predominant throughout the 30 year history of commercial wind turbines. The most essential reason for the unpopularity of the downwind turbines is the presence of a strong aerodynamic interaction between the rotor and the tower, which is commonly known as the “tower shadow effect”. This phenomenon generates impulsive loads and infrasound when one of the blades passes through the wake of the tower [1]. The modeling of the tower shadow effect is a critical technical problem of downwind turbines.

Design loads are calculated on the basis of the international design standard IEC61400-1 [2] or the guidelines for certification bodies DNV GL [3]. In a large number of cases in which the design load is combined with the wind model, the wind turbines can experience failure conditions, as well as various types of wind and marine conditions. The flexibilities of the structure and the controls, in addition to the aerodynamics, hydrodynamics, aero-elastics, and control of the wind turbines, considerably influence the load. The blade element momentum (BEM) method is the most popular tool for determining these characteristics [4,5]. Therefore, the modeling of the tower shadow effect in the BEM is one of the most important technical challenges in the design and development of downwind wind turbines. However, most of the tower shadow models are too simple to express such phenomena, as they consider only the wind speed profile behind an isolated tower [5,6] and not the aerodynamic interaction between the rotor and the tower.

Several studies have been conducted until now that have focused on the variable loads of the downwind turbines caused by the tower shadow effect. Zhao et al. [7] studied the loads on a tower by performing a comparison between two- and three-bladed upwind and downwind rotors in two different rotor speed conditions using computational fluid dynamics (CFD). The peak to average values of the hub bending moment and the tower base bending moment of the downwind turbine were shown to be 5–6 times and 2.5 times larger, respectively, than those of the upwind turbine. Zahle et al. [8] conducted 2-dimensional CFD analysis for three typical tower concepts of downwind turbines as well as for an upwind turbine; a steep load variation was noted for the downwind turbine, and the implementation of streamlined and four-leg towers was shown to be an effective technique to reduce the load variation caused by the tower shadow effects. Matiz-Chicacausa and Lopez [9] investigated the possibilities of the application of the actuator line model (ALM) to the CFD analysis of the tower shadow effects. The ALM was shown to be useful for load estimation of a downwind turbine under specific conditions.

Although the abovementioned research contributed to the expansion of the understanding of the phenomenon, the authors did not aim to develop models for extensive load calculations based on the design standards and guidelines. Wang and Coton [10] developed a high-resolution tower shadow model for downwind turbines based on the prescribed wake vortex model and an efficient near wake dynamics model of the vorticity trailing the blades. The model demonstrated reasonable agreement with the experimental results, with the exception of the blades passing through the tower wake. However, the load of the tower was not discussed in this research. Yoshida and Kiyoki [11,12] developed the load equivalent tower shadow modeling method. The method considered a bell-shaped wind speed profile behind the tower, which helped obtain the equivalent rotor load variation with the wind turbine CFD analysis. This model was applied to the development of the first multimegawatt-scale commercial downwind turbines, including the SUBARU 80/20 [13] and, later, the Hitachi 2 MW, to realize high performance and safety in complex terrains [14,15]. The technology was extended to the later produced Hitachi 5 MW [16] offshore wind turbines. The model has been successfully applied in engineering applications so far; however, two major problems remained unsolved. The first concern was that the CFD modeling of the rotor–tower configuration required for each operating condition was time-consuming, which made the technique inconvenient for use in the design optimization process. The second issue was that no model was established for the tower shadow effects on the tower drag. Considering the situation, the tower load variation caused by the tower shadow effect was modeled by Yoshida by combining the BEM and the Lifting Line Theory [17]. However, the average tower drag was still not discussed in that research.

Considering these situations, this paper proposes a BEM model for the tower average drag. The formulation is presented in Section 2, and the validation of the method by a wind tunnel test is discussed in Section 3, Section 4 and Section 5.

2. Methodology

The formulation of the average loads of the towers of the downwind turbine by using the BEM is discussed in this chapter. The following two calculations are considered:

(1)

Wind speed distribution by CFD

(2)

Rotor thrust-induced tower average drag

2.1. Wind Speed Distribution by CFD

The wind speed distribution in front of the rotor is calculated using CFD. Figure 1 shows the top view of the rotor and the tower of a downwind turbine. The tower, which has the diameter D_T, is located in front of the rotor. The free stream wind speed U₀ decreases to U₀(1 − a) at the rotor plane as a result of the rotor thrust. Here, a is the axial induction factor. Figure 2 is the schematic of the CFD involved in the present study. The tower is not included in the model, but the position is termed “virtual tower position” in this research. Further, u_T is the wind speed at the virtual tower position, which is between U₀ and U₀(1 − a), depending on the condition. The rotor is modeled by an actuator disc model (ADM). The load distribution is assumed to be uniform to realize a convenient application to the BEM. The thrust per unit volume σ of the ADM is as shown in Equation (1):

$$\sigma =\frac{1}{2\pi rw}\frac{dT}{dr}$$

(1)

where w is the thickness of the ADM.

2.2. Rotor Thrust-Induced Tower Average Drag

The wind speed at the tower and the tower drag decrease as the rotor thrust increases due to the aerodynamic interaction. Two factors, that is, the wind speed and the ambient pressure gradient caused by the rotor thrust, are considered in this research. The formulation is based on the assumption of the strip theory, according to which the flow between the sections normal to the tower axis is not taken into account.

(1) Wind Speed Effect

The tower drag decreases proportionally to the square of the wind speed when the tower section drag coefficient remains constant. This influence is modeled herein.

The section drag f_XT0 of the isolated tower with no rotor interaction is as shown in Equation (2):

$$f_{XT0}=\frac{1}{2}\rho U_0{}^2{D}_T{C}_{dT0}$$

(2)

where C_dT0 is the drag coefficient at the tower section.

In case the effect of the rotor thrust comes into play, the wind speed at the tower position decreases. Furthermore, the tower section drag also decreases from f_XT0 to f_XT. This is expressed in the following two ways, based on the wind speeds at the free stream and at the virtual tower position, as shown in Equation (3):

$$f_{XT}=\frac{1}{2}\rho U_0{}^2{D}_T{C}_{dT}=\frac{1}{2}\rho u_T{}^2{D}_T{C}_{dT0}$$

(3)

where C_dT is the tower section drag based on the free stream wind speed, and u_T is the wind speed at the virtual tower position, which is calculated by CFD without incorporating a tower model, as explained in the previous section. Here, the drag coefficient C_dT0 is assumed to be constant as in Equation (2).

Therefore, the tower section drag deviation induced by the rotor thrust Δf_XT is calculated as in Equation (4):

$$\Delta f_{XTV}=f_{XT}-f_{XT0}=-\frac{1}{2}\rho U_0{}^2{D}_T{C}_{dT0}\left(1-{\mu }_T{}^2\right)$$

(2)

where the normalized wind speed at the virtual tower is

$${\mu }_T=\frac{u_T}{U_0}$$

Therefore, the change in the relevant drag coefficient caused by the wind speed change ΔC_dTV is calculated by Equation (4), as given in Equation (5):

$$\Delta C_{dTV}=-C_{dT0}\left(1-{\mu }_T{}^2\right)$$

(3)

This indicates that the term of the tower section drag decreases as the normalized wind speed at the virtual tower, induced by the rotor thrust, decreases.

(2) Effect of the Ambient Pressure Gradient

The tower drag is also dependent on the ambient pressure gradient around the tower. This influence is modeled herein.

The pressure at the virtual tower position p_T is calculated by Bernoulli’s law, as in Equation (4):

$$p_0+\frac{1}{2}\rho U_0{}^2=p_T+\frac{1}{2}\rho u_T{}^2$$

(4)

where p₀ and p_T are the pressures at the free stream and the virtual tower center, respectively. Therefore, the windward pressure gradient ∂p_T/∂x_T at the virtual tower center is calculated as shown in Equation (5) by the differential of Equation (4):

$$\frac{\partial p_T}{\partial x_T}=-\rho u_T\frac{\partial u_T}{\partial x_T}$$

(5)

The section drag by the pressure gradient at the tower section Δf_XTP is calculated as in Equation (6) by employing the pressure deviation from the front and back sides of the tower and the tower diameter. The pressure deviation is calculated by the pressure gradient and the windward reference distance $\Delta x_T$:

$$\Delta f_{XTP}=\frac{\partial p_T}{\partial x_T}\Delta x_T{D}_T$$

(6)

Assuming a uniform pressure gradient, the tower section drag is expressed as in Equation (7):

$$\Delta f_{XTP}={\displaystyle {\int }_{-\pi /2}^{\pi /2}\left(\frac{\partial p_T}{\partial x_T}{D}_T\cos {\varphi }_T\right)\left(\frac{D_T}{2}\cos {\varphi }_T\right)d{\varphi }_T}$$

(7)

The reference distance is calculated as in Equation (8) using Equations (6) and (7):

$$\Delta x_T=\frac{\pi }{4}{D}_T$$

(8)

Therefore, the change in the tower section drag owing to the pressure gradient is calculated as in Equation (9):

$$\Delta f_{XTP}=\frac{\pi D_T{}^2\rho u_T}{4}\frac{\partial u_T}{\partial x_T}=\frac{\pi D_T\rho U_0{}^2{\mu }_T}{4}\frac{\partial {\mu }_T}{\partial {\xi }_T}$$

(9)

where ξ_T is the normalized distance and can be written as

$${\xi }_T=\frac{x_T}{D_T}$$

Therefore, the change in the relevant drag coefficient caused by the pressure gradient ΔC_dTP can be written as in Equation (10):

$$\Delta C_{dTP}=\frac{\pi }{2}{\mu }_T\frac{\partial {\mu }_T}{\partial {\xi }_T}$$

(10)

This indicates that the term of the tower section drag decreases as the negative pressure gradient caused by the rotor thrust increases.

(3) Total Average Tower Drag

From (1) and (2), the deviation of the drag Δf_XT and the drag coefficient ΔC_dT induced by the rotor thrust are derived as Equations (11) and (12), respectively:

$$\Delta f_{XT}=\Delta f_{XTV}+\Delta f_{XTP}=\frac{1}{2}\rho U_0{}^2{D}_T\left[-C_{dT0}\left(1-{\mu }_T{}^2\right)+\frac{\pi }{2}{\mu }_T\frac{\partial {\mu }_T}{\partial {\xi }_T}\right]$$

(11)

$$\Delta C_{dT}=\Delta C_{dTV}+\Delta C_{dTP}=-C_{dT0}\left(1-{\mu }_T{}^2\right)+\frac{\pi }{2}{\mu }_T\frac{\partial {\mu }_T}{\partial {\xi }_T}$$

(12)

3. Wind Tunnel Test

3.1. Outline

A wind tunnel test for a wind turbine was conducted to validate the theory described in the previous chapter. The rotor–tower interaction was simulated by a dummy tower placed in front of the rotor of the upwind turbine.

3.2. Facility

The Boundary Layer Wind Tunnel at the Research Institute for Applied Mechanics, Kyushu University [18], was used for the test. The wind tunnel has a test section with a width of 3.6 m, a height of 2.0 m, and a length of 15 m. The maximum wind speed is 30 m/s.

3.3. Test Model

The general specifications and the outline of the model and the tower installed in the wind tunnel are presented in

Table 1. Specifications of the test model and the dummy tower.

Item	Specification
Number of blades	2
Rotor diameter D	700 mm
Rotor radius R	350 mm
Tilt angle	0 deg
Coning angles	0 deg
Airfoil	NACA0013
Blade chord length	32 mm
Twist angle	0 deg
Tower diameter DT	64 mm
Tower position	−338 mm (−6DT) −225 mm (−4DT)

. The blockage ratio, which is the ratio of the rotor area to the cross section of the wind tunnel, is as small as 5.3%. The rotor speed is controlled by the variable dump load. Figure 3 shows the schematic of the test model and the dummy tower. Although the model is an upwind turbine supported by a tower with a diameter of 50.8 mm, a dummy tower is installed in front of the model to mimic the downwind turbine tower. The diameter of the dummy tower is 64 mm, which is 200% the value of the blade chord length. The dummy tower was placed at two positions; −6D_T and −4D_T. Figure 4 shows an image of the test model. The dummy tower is shown as located in front of the turbine.

3.4. Measurement

-

Rotor speed: Magnetic encoder

-

Load (top of the tower): 6-axis Load Cell (Nissho)

-

Pressure on the dummy tower (50%R, 80%R, 8 points each): Pressure sensor (Otegiken)

3.5. Test Conditions

-

Wind speed (U₀): 6 m/s (uniform, steady)

-

Yaw angle: 0 deg

-

Blade pitch angle (θ): 4, 6, 8 deg

-

Tip speed ratio (λ): 6.6–9.2

3.6. Test Results

The measurements for the power coefficient C_P and the thrust coefficient C_T for the isolated rotor model are shown in Figure 5. The rotor torque was calculated from the rolling moment at the top of the tower. The value of thrust force at the top of the tower was used as the rotor thrust. The maximum value of C_P was noted at approximately λ = 8 and θ = 6 deg. However, C_T tended to increase with an increase in the tip speed ratio or a decrease in the pitch angle.

The pressure distributions on the dummy tower in typical cases are shown in Figure 6. Here, 0 deg indicates the upwind side of the tower. The pressure on the 50%R was relatively higher than that at 80%R. In general, the pressures around the downwind side of the tower (180 deg) tended to be higher as the rotor thrust was larger.

The details of other characteristics are discussed in Section 5.

4. Analysis

4.1. Outline

The relationship between the rotor thrust and the tower drag coefficient was determined by the proposed method described in Section 2. The model is generally the same as the one described in Section 3.3; however, the influence of the support was neglected as the support was located far downwind of the rotor, and this distance was fairly large compared to the support’s diameter. In addition, the influence of the nacelle was neglected, as it does not affect the outboard sections, as discussed in this study.

4.2. CFD Outline

A CFD analysis was conducted for the rotor using ANSYS CFX [19] considering the k-ω SST turbulence model. The rotor was modeled by an ADM with uniform load distribution.

The simulation domain and the boundary conditions are summarized in

Table 2. Simulation domain and boundary conditions.

Boundary	Distance from the Rotor Center	Boundary Condition
Rotor	−D/140~+D/140	ADM
Tower	Not available	Not available
Inlet	−4D	Wind speed (uniform)
Outlet	+5D	Ambient pressure
Bottom	−2D	Slip
Top	+2D	Slip
Side	−2D, +2D	Slip

“D”: Rotor diameter.

. The model consisted of structured cells around the ADM with y+ is about 1 and unstructured ones otherwise. The total number of cells was approximately 50 million. The typical cells are shown in Figure 7 and Figure 8.

4.3. Analysis Conditions

-

Wind speed (U₀): 6 m/s (uniform, steady)

-

Yaw angle: 0 deg

-

Thrust coefficient (C_T): 0.4–0.9 (each 1.0)

4.4. Wind Speed Distributions in Front of the Rotor

The wind speed distributions in typical conditions, C_T = 0.4 and 0.9, are shown in Figure 9 and Figure 10. The circles in these figures denote the positions of the dummy towers at −6D_T and −4D_T. The wind speed tended to decrease in front of the tower in general, and the wind speed was lower in the vicinity of the rotor. A comparison between the two conditions shows that the wind speed around the dummy tower decreased as C_T increased. A comparison between the two sections shows that the wind speed in front of the rotor was lower at 50%R compared to that at 80%R. The load was uniform on the rotor, and the inboard sections were considerably affected by the rotor.

The distributions of the normalized wind speed μ_T and its differential ∂μ_T/∂ξ_T with respect to the normalized distance ξ_T in front of the rotor are shown in Figure 11 and Figure 12. The distributions at C_T = 0.4 and 0.9were identical to those in Figure 9 and Figure 10 in the symmetrical plane. The top subplots correspond to the distributions at 50%R, and the bottom ones correspond to those at 80%R. The wind speed was lower in the vicinity of the rotor and decreased as the thrust coefficient increased, as mentioned above.

4.5. Drag Coefficients of the Virtual Tower

The deviation of the drag coefficient of the virtual tower calculated by the present theory is shown in Figure 13. The x-axis denotes the virtual tower position, and the y-axis denotes the thrust coefficient. The diameter of the tower D_T was assumed to be 64 mm, as in the experiment. As indicated by Figure 11 and Figure 12, the drag coefficient tended to be smaller as the rotor–tower clearance decreased and the thrust coefficient increased. The drag tended to be smaller at 50%R than at 80%R.

As shown in Equation (14), the rotor thrust-induced tower drag deviation consisted of the wind speed term and the pressure gradient term. Figure 14 shows the share of the wind speed term to the total deviation. It indicates that approximately 80% of the change in the drag wss caused by the first term of Equation (12), i.e., the decrease in the wind speed, rather than the pressure gradient.

5. Validation

The thrust-induced drag deviations ΔC_dT to the thrust coefficient at −6D_T and −4D_T are shown in Figure 15. The lines are approximations for which the intercepts are at ΔC_dT = 0 and C_T = 0 by definition. The ΔC_dT values for all the cases are shown to be almost proportional to C_T. The proportion factors are summarized in

Table 3. Fraction of the deviation of the tower drag coefficient to the thrust coefficient.

Dummy Tower Position	Tower Section	Test	Present
		ΔCdT/CT [-]	ΔCdT/CT [-]	Deviation from the Test
−6DT	50%R	−0.1270	−0.0937	−26.2%
	80%R	−0.1058	−0.0741	−30.0%
−4DT	50%R	−0.1484	−0.1594	+7.4%
	80%R	−0.0961	−0.1225	+27.5%

. The factors of the former BEM are zero, as the method does not consider the influence of the rotor thrust on the tower drag. On the other hand, the proposed method shows good agreement with the test. The drag coefficients tended to decrease as the rotor thrust increased. The drag coefficient at the 50%R section was smaller than at the 80%R section. Further, the drag coefficient decreased as the tower was placed closer to the rotor.

6. Conclusions

A novel analysis method to calculate the average tower drag of downwind turbines, which considers the rotor-induced average tower drag coefficient, was developed. It consists of two terms, that is, the decrease in the wind speed and the pressure gradient caused by the rotor thrust. The rotor thrust distribution is assumed to be uniform. The method was validated by a wind tunnel test. Unlike the former blade element momentum (BEM) method, which assumes the tower drag to be constant, the proposed method demonstrates a much better agreement with the wind tunnel test, with an accuracy of up to 30%. The results show that the tower drag decreases proportionally to the rotor thrust. With regard to the sensitivity of the tower drag with respect to the rotor thrust, the following characteristics were noted:

-

The drag coefficient was larger in the inboard section (50%R) than in the outboard section (80%R).

-

The drag coefficient decreased as the tower was positioned closer to the rotor.

-

Of the two terms influencing the deviation of the tower section drag, the effect of the decrease in wind speed was more dominant in leading to the decrease in the tower section drag.

By preparing the database of the relationship between the rotor thrust coefficient for the wind speed distribution in front of the rotor, the proposed method is expected to improve the accuracy of the load calculation in the BEM.

In future studies, the present model is planned to be extended to variable loads of wind turbine blades in succession.