Wind tunnel test on aerodynamic coefficients of multi-bundled conductors under skew winds

Journal of Fluids and Structures 91 (2019) 102702

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Wind tunnel test on aerodynamic coefficients of multi-bundled conductors under skew winds ∗

Hai Jiang Xu a , Hong Zhou Deng a , , Xiao Yi Hu a , Xing Ma b a b

Department of Structural Engineering, Tongji University, Shanghai, 200092, China School of Natural and Built Environments, University of South Australia, Adelaide, Australia

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Article history: Received 9 January 2019 Received in revised form 7 July 2019 Accepted 9 August 2019 Available online xxxx Keywords: Skew wind Aerodynamic coefficients Multi-bundled conductors Wind tunnel test Independence principle Shielding effect

a b s t r a c t A series of wind tunnel tests were conducted to study the aerodynamic coefficients of multi-bundled conductors under skew winds. Inclined model conductors were designed to simulate various wind incidence angles. Several coefficients and factors were defined for analyzing the effect of the Reynolds number, the angle of attack, the shielding effect, and the wind incidence angle on the aerodynamic coefficients. The test results were validated using available data for a wind incidence angle of 90◦ . It was found that the drag coefficient-Reynolds number curve of the conductors differed considerably from that of a circular cylinder. A value of 1.0 is recommended for the drag coefficient of a single conductor. The aerodynamic coefficients were strongly dependent on the angle of attack, and a change in the angle of attack resulted in the aerodynamic coefficients varying by more than 10%. Furthermore, it was observed that the shielding effect can lead to the overestimation of the wind load by up to 25%. A value of 0.95 is recommended for the shielding effect coefficient. The independence principle can be reliably used for practical design, even though it causes the normal wind load to be slightly overestimated. Axial wind loads for wind incidence angles of 75◦ and 45◦ were remarkably high. It is recommended that for these angles, they be considered to be 0.3 and 0.55 times the normal wind load, respectively. Finally, a simple method for calculating the normal and axial force coefficients under skew winds is proposed. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction With the increasing popularity of ultra-high voltage transmission lines, the traditional single conductor is gradually being replaced by multi-bundled conductors because of the superiority of the latter in suppressing corona discharge and reducing the line reactance. Multi-bundled conductors comprise a set of closely spaced traditional conductors (each one is called a sub-conductor) that are parallelly arranged in the form of polygons. Transmission lines are wind-sensitive, and therefore, the accurate calculation of the wind load on them, especially on the aerodynamic coefficients of multi-bundled conductors, is very important. For the practical arrangement of conductors against the wind, two angles should be carefully considered: the angle of attack, which is the global rotation angle of multi-bundled conductors about their own axis, and the wind incidence angle, which is the angle between the conductor axis and the wind direction. Since the 1960s, the high-frequency force balance (HFFB) wind tunnel test has been the most preferred and effective method to study the aerodynamic coefficients of conductors (Counihan, 1963). On the basis of a test performed in the ∗ Corresponding author. E-mail address: [email protected] (H.Z. Deng). https://doi.org/10.1016/j.jfluidstructs.2019.102702 0889-9746/© 2019 Elsevier Ltd. All rights reserved.

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Nomenclature CD , CL Cn , Cp CX , CY , CZ d0 , d D, L Fi l n, p N Re SF SEC U WAAC WIAC X, Y , Z

ρ φ θ

Drag coefficient and lift coefficient, respectively Normal force coefficient and axial force coefficient, respectively Force coefficients in the X , Y , and Z directions, respectively Strand diameter and conductor diameter, respectively Wind axes in the along-wind and across-wind directions, respectively Wind load in the i direction Conductor length Body axes in the directions normal and parallel to conductor axis, respectively Bundle number Reynolds number Scale factor Shielding effect coefficient Wind speed Angle of attack coefficient Wind incidence angle coefficient High-frequency force balance axes Air density Wind incidence angle Angle of attack

Subscripts 1 i max, min

The value of a single conductor The value about the i-axis (i = D, L, X , Y , Z , n, p) Maximum and minimum values, respectively

open air with a tunnel-like test setup, a previous study showed that the results of the HFFB wind tunnel test agreed well with those of full-scale field measurements (Shan et al., 1992). Past studies involving HFFB tests on conductors have mainly focused on three categories of influencing factors: (i) the surface roughness and turbulence intensity, which have similar effects, (ii) the sub-conductor arrangement in bundles, including the number of bundles, spacing between bundles, and angle of attack, and (iii) inclined-flow conditions or the wind incidence angle. Achenbach (1971) performed a wind tunnel test on a circular cylinder with various surface roughness and found that in the postcritical region, a higher surface roughness corresponded to a lower critical Reynolds number (Re) and a higher drag coefficient. Similarly, it has been shown that a conductor has a considerably lower critical Re and a higher drag coefficient than a common circular cylinder (Wang et al., 2016; Wardlaw et al., 1975; Xie et al., 2011; Zuo et al., 2015). Such studies provide ideas for the design of new types of conductors with different types of surfaces for reducing the wind load (Eguchi et al., 2002). In particular, the effect of turbulence is controversial; while some tests have shown that a higher turbulence intensity corresponds to a higher drag coefficient (Lou et al., 2015; Zuo et al., 2015), others have drawn the opposite conclusion (Xie and Guan, 2013). Research on the interference effect of a cluster of tall buildings dates back to the 1930s (Harris, 1934). The term interference effect has a broad range of meanings and refers to all types of influence that an upstream structure has on a downstream structure, including the wind load, dynamic characteristics, aerodynamic characteristics, and wind-induced responses of the downstream structure. In the present paper, the term interference effect is used to convey a narrow range of meanings. The influence of an upstream conductor on the aerodynamic coefficients of a conductor in a downstream conductor bundle is termed shielding effect. The shielding effect is related to many parameters, such as the number of bundles, spacing between bundles, and the angle of attack. Chinese scholars have conducted many studies on the shielding effect (Lou et al., 2015; Sun and Xie, 2013; Wang et al., 2016; Xie et al., 2011, 2013b; Xie and Guan, 2013; Zuo et al., 2015), and it has been found that it is negatively correlated with the bundle spacing and positively correlated with the conductor diameter and bundle number. Furthermore, it is strongly dependent on the angle of attack. Although, the shielding effect may lead to a reduction in the wind load in excess of 10%, it is not considered in many codes such as the Chinese code (DL/T 5154-2012, 2013). It should be emphasized that in past studies, HFFB tests on multi-bundled conductors have been performed for normal wind incidence (i.e., for a wind direction normal to the conductor axis). In the practical design of transmission lines, the parameters that are of the most interest are the axial forces of tower members. Wind tunnel tests have shown that axial forces reach a maximum at a wind incidence angle of 45◦ or 60◦ (Xie et al., 2013a). Therefore, skew wind conditions should be considered in the practical design of transmission lines. Generally, the normal wind load on cylinders is calculated using the independence principle. The assumption that flow

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Fig. 1. (a) Schematic cross-sectional view and (b) spiral angle of the prototype conductor.

around an inclined circular cylinder is quasi-two-dimensional is implicit in this principle (Hayashi and Kawamura, 1995). In other words, the near-wake region is dominated by vortex structures aligned parallel to the cylinder. Consequently, the forces normal to an inclined cylinder are the same as those normal to a non-inclined cylinder using only the component of the free-stream flow normal to the cylinder axis (Marshall, 2003). The independence principle has been proved to be valid for an inclined cylinder as well as two parallel inclined cylinders at a large or moderate wind incidence angle (Thakur et al., 2004). However, it is not accurate for inclined finite-aspect-ratio cylinders at wind incidence angles smaller than 45◦ (Vakil and Green, 2009). For inclined conductors, test results show that the independence principle is valid in the subcritical region and less accurate in the supercritical region (Macdonald and Larose, 2006). Studies on inclined multi-bundled conductors are scarce. In view of the influence of the wind incidence angle on the shielding effect, the applicability of the independence principle to multi-bundled conductors under skew winds requires validation. Previous research has mostly focused on the drag coefficient of a single conductor under skew winds and of multibundled conductors at a wind incidence angle of 90◦ . In particular, investigations of the drag coefficient of multi-bundled conductors under skew winds have been scarce. Moreover, the axial forces of conductors have received little attention. Therefore, in the present study, the HFFB test was conducted on multi-bundled conductors at wind incidence angles of 90◦ , 75◦ , 60◦ , and 45◦ . The effect of the Reynolds number, the angle of attack, the shielding effect, and the wind incidence angle on the normal and axial force coefficients of multi-bundled conductors under skew winds was investigated. Furthermore, a simple method for calculating the aerodynamic coefficients is proposed. 2. Wind tunnel test 2.1. Prototype and model conductors The prototype conductor used in the present study was a type of steel-reinforced aluminum cable (ACSR) referred to as JL/G1A-630/45. It has been used in many previous studies (Wang et al., 2016; Xie et al., 2013b; Zuo et al., 2015), and therefore, it was considered appropriate for validating the present test results. The conductor consisted of seven steel strands at the center enclosed by three concentric layers of aluminum strands, as shown in Fig. 1(a); in the figure, d is the conductor diameter and d0 is the diameter of the outermost strand. As stipulated by the Chinese code (GB/T 1179-2008, 2009), the strands were twisted around the conductor axis at a spiral angle (i.e., the inclination angle between the strand axis and the horizontal plane) of 74.1◦ (Fig. 1(b)). Owing to its low stiffness, the prototype conductor is not suitable for the HFFB test. In the absence of a special treatment such as prestressing (Stroman, 1997), the position of its axis can change easily under the conditions of the HFFB test. Therefore, in the present study, a model conductor was used as an alternative. The design of a model conductor should satisfy two conditions: (i) The model conductor should be aerodynamically similar to the prototype conductor. This implies that the principal aerodynamic shape parameters (conductor diameter, strand diameter, and spiral angle) of both conductors should be identical. (ii) The model conductor should have high stiffness to prevent resonance. An aluminum tube was used as the core to achieve high stiffness, and a layer of rubber strands, which was concentric with and enclosed the aluminum tube, was used to simulate the aerodynamic shape of the prototype conductor (Fig. 2). These materials were also used by Xie et al. (2013b) and Zuo et al. (2015).

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Fig. 2. Schematic cross-sectional view of the model conductor.

2.2. Test setup As shown in Fig. 3(a) and (b), the test setup was vertically installed on a turntable in the wind tunnel for facilitating the control of the angle of attack. The main components of the test setup were the top end-plate, rectangular plate, bottom end-plate, HFFB, and strut. A photograph of the top end-plate is shown in Fig. 3(c). The end plate was a circular plastic plate glued to the top of a model conductor, and it was used to ensure that the incoming airflow was two-dimensional (2D). The rectangular plate was welded to the base of the conductor (Fig. 3(d)), and it was affixed to the bottom end-plate by bolts. The wind direction was always normal to its leading edge. The windward face was beveled for smooth airflow; in other words, it was ensured that flow separation was weak. The bottom end-plate was a large circular aluminum alloy plate. It was hollowed out and beveled in order to reduce the self-weight, vertical vibrations, and flow separation. The bolt holes corresponded to all the possible arrangements of the multi-bundled conductors (Fig. 3(a)). The diameter of the bottom end-plate was determined by the spacing between adjacent conductors, which in turn was determined by the wind incidence angle. An HFFB with six degrees of freedom was installed beneath the bottom end-plate, and a photograph of the HFFB is shown in Fig. 3(e). Its resolution was 0.0625 N in the X and Y directions and 0.125 N in the Z direction. The aluminum strut shown in Fig. 3(f) was designed to separate the test setup above the HFFB from the boundary layer created at the bottom of the wind tunnel. Xie et al. (2013b), who used the same wind tunnel, measured the thickness of the boundary layer to be about 0.5–0.6 m for a turbulence intensity of 7%. The heights of the strut and HFFB (about 450 mm) were sufficient to separate the test setup from the boundary layer because the thickness of boundary layer was smaller under the present test conditions; in other words, the turbulence intensity was almost zero. It is desirable to restrict the movement of both ends of a conductor to meet the stiffness requirement, as done by Xie et al. (2013b). However, under skew wind conditions, when the angle of attack changes, the conductor axis precesses and the position of the top end of the conductor changes (Fig. 4). This warrants the adjustment of the position of the apparatus used to restrain the top end of conductors (usually a large end-plate similar to bottom end-plate is used), which is rather time-consuming. For convenience, during the test operation, only the bottom end of the conductor was restrained. As shown in Figs. 3(a) and 4, the conductor was inclined to set the desired wind incidence angle. In the present study, the inclination angle of each conductor equaled the wind incidence angle, and the conductor axis was always parallel to the side walls of the wind tunnel. Whenever the angle of attack changed and the conductor axis moved from the solid line to the dash line in Fig. 4, the wind incidence angle changed as well. Each conductor had to be moved to its original position (solid line) by loosening, rotating, and tightening it step by step. Since the bottom end-plate was not rotated, the movement of the conductor did not change the angle of attack. The wind incidence angle and angle of attack were independent of each other. 2.3. Test conditions The wind tunnel tests were conducted in the TJ-2 wind tunnel at Tongji University. The test section dimensions were 2.5 m × 3 m × 15 m (height, width, and length, respectively). A turntable with a diameter of 1.85 m was positioned near one end of the test section.

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Fig. 3. (a) Sketch map of the test setup (b) test setup in the wind tunnel, and components of the test setup including (c) top end-plate (d) rectangular plate (e) HFFB, and (f) strut.

Fig. 4. Motion track of a conductor’s top end under skew winds when the angle of attack changes.

The tests were conducted under uniform flow for wind incidence angles of 90◦ , 75◦ , 60◦ , and 45◦ (Fig. 5). The calibrated turbulence intensity was less than 0.1%, and the conductor length l corresponding to the wind incidence angle φ was 500/sinφ . For each wind incidence angle, a single conductor was tested at first to examine the effect of the Reynolds number. Subsequently, four-, six-, and eight-bundled conductors were tested. The angles of attack for the multi-bundled conductors are shown in Fig. 6. They were chosen on the basis of the planar symmetry of the bundle and Xie et al.’s (2013b) values of the maximum and minimum aerodynamic coefficients. The bundle spacing was 400 mm, which is the optimal value for practical design (He et al., 2012). The wind speed U was mostly 5, 10, 15, 20, and 25 m/s for the test

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Fig. 5. Definition of wind incidence angle. The vertical projected length of a model conductor was 500 mm. The wind incidence angle equaled the inclination angle of a model conductor.

Fig. 6. Angles of attack for (a) four-bundled conductors (b) six-bundled conductors, and (c) eight-bundled conductors used in the HFFB tests.

conditions, except for the wind incidence angle of 45◦ , for which 22 m/s was the maximum wind speed to avoid vertical vibrations in the bottom end-plate. The sampling frequency of the HFFB was 300 Hz, and data were collected at 60 s intervals in each test run after the wind speed was stabilized. 2.4. Definitions of coefficients and factors The coordinate systems used in the present study are shown in Fig. 7. U indicates the wind direction, and the X , Y , and Z axes are termed HFFB axes. The Z -axis is perpendicular to the horizontal XY plane. The D and L axes are termed wind axes, and they represent the along-wind and across-wind directions, respectively. The DL plane is coplanar with the XY plane. The n and p axes are referred to as body axes. They are perpendicular and parallel to the conductor axis, respectively, and are defined on the basis of the wind incidence angle. The np plane is always parallel to the DZ plane as well as wind tunnel walls. When the angle of attack θ changes, the n, p, Z , D, and L axes remain unchanged, while the X and Y axes rotate anticlockwise about the Z -axis. As shown in Fig. 5, θ is the angle between the D and L axes. 2.4.1. Aerodynamic coefficients The data obtained in the HFFB test comprised six components, namely, the forces (FX , FY , and FZ ) and moments (MX , MY , and MZ ) about the HFFB axes. During the test, the forces acting on the bottom end-plates, Fi,plate (i = X , Y , Z , D, L, n, p), were measured under test conditions (U, φ , and θ ) identical to those during the measurement of the forces in the presence of conductors, Fi,plate+conductors , to obtain the net forces (given by Fi = Fi,plate+conductors − Fi,plate ) acting on the conductors.

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Fig. 7. Coordinate systems used in the present study.

However, the net moments acting on the conductors cannot be calculated similarly because they are related to the centroid of Fi,plate and Fi,plate+conductors , or the (nonuniform) distribution of the mass of the bottom end-plate and forces along the conductor axis. Therefore, the moments were not studied. The aerodynamic coefficient in the i direction is defined as 1 Ci = Fi /( ρ U 2 dlN) (1) 2 where ρ is the air density, N is the bundle number, and CD , CL , CX , CY , CZ , Cn , and Cp are the drag coefficient, lift coefficient, force coefficient about X -axis, force coefficient about Y -axis, vertical force coefficient, normal force coefficient, and axial force coefficient, respectively. The interrelationships among these coefficients (Fig. 5) can be expressed as

⎧ C = CX cos θ − CY sin θ ⎪ ⎨ D CL = CX sin θ + CY cos θ ⎪ ⎩ Cn = CD sin φ + CZ cos φ Cp = CD cos φ − CZ sin φ

(2)

The lift coefficient was approximately zero for all test conditions; therefore, it can be neglected in practical design. The blockage ratio of model conductors and the test setup ranged only up to 2%. According to the Japan Research Committee of Guide for Wind Tunnel Experiment (2011), the aerodynamic coefficients need not be corrected. 2.4.2. Angle-of-attack coefficient During the HFFB test, in which different angles of attack were considered for a specific wind speed and wind incidence angle, the maximum and minimum values of the aerodynamic coefficient were recorded as Ci,max and Ci,min , respectively. The angle of attack coefficient WAAC was then defined as the ratio of the variation of Ci (i.e., Ci,max − Ci,min ) to Ci,min : WAACi =

Ci,max − Ci,min Ci,min

× 100%

(3)

2.4.3. Shielding effect coefficient The shielding effect was represented by the difference between the average aerodynamic coefficient of multi-bundled conductors, Ci , and that of a single unshielded conductor, Ci,1 , under the same test conditions (U and φ ). From Eq. (1), we can write Ci,1 = Ci (N = 1). The shielding effect coefficient SEC was defined as the ratio of Ci to Ci,1 . The parameter Ci , which lies in the range Ci,min − Ci,max , corresponds to SEC i , which lies in the range SEC i,min –SEC i,max .

⎧ Ci,max ⎪ ⎨ SECi,max = Ci,1

C ⎪ ⎩ SECi,min = i,min

(4)

Ci,1

2.4.4. Wind incidence angle coefficient The influence of the wind incidence angle can be represented by the difference between the normal force coefficient at a specific φ , Cn (φ ), and the normal force coefficient at φ = 90◦ , Cn (90◦ ), under the same test conditions (U and θ ). The wind incidence angle coefficient WIAC n was then defined as the ratio of Cn (φ ) to Cn (90◦ ). WIACn =

C n (φ ) Cn (90◦ )

(5)

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Fig. 8. Appearance quality of model conductors: (a) worst quality and (b) best quality. Table 1 Comparison of the drag coefficients of model conductors with different appearance quality at U = 10 m/s. Appearance quality

CD

Best Worst

1.195 1.134

2.4.5. Scale factor The easiest way to calculate the axial wind load on conductors is to multiply the normal wind load by a factor, namely, scale factor (represented by the variable SF ). It is defined as the ratio of the axial force coefficient Cp to the normal force coefficient Cn under the same test conditions (U, φ , and θ ). SF =

Cp Cn

(6)

3. Results and discussion 3.1. Discussion of model conductor and test setup In the design stage, some simplifications were made to the model conductor and test setup. This section discusses the effect of the simplifications on the results obtained. First, the winding of strands was manual. Therefore, apart from residual glue on strands, there were gaps between strands (Fig. 8). The drag coefficients of conductors with the worst and best appearance quality are compared in Table 1. The difference is clearly less than 5%, indicating that the test results are affected only to a small extent by the appearance quality of the model conductors. In all the tests in this study, models with the worst appearance quality were excluded. Second, because of the difficulty in installing large top end-plates (of the same size as a bottom end-plate, which is shown in Fig. 3(a)) in the wind tunnel and for simplifying the test operation, small top end-plates (Fig. 3(c)) were used; such a small top end-plate was also used by Belloli et al. (2010). In addition, tests were conducted under other top-end conditions, namely, without any cover at the top and with an end cover, as shown in Fig. 9. In the absence of a cover, the top of the conductors is completely open and airflow enters the tube. The end cover is used to ensure that the tube is sealed. However, the flow may separate at the leading edge of the end cover, leading to a flow component in the Z -direction. All these top-end conditions cannot generate 2D flow, and the flow is three-dimensional (3D). As shown in Table 2, the models with a top end-plate had a drag coefficient that was at least 10% greater than the drag coefficients for the other two conditions. A similar difference in the drag coefficients was observed in the studies of Xie and Guan (2013) and Xie et al. (2013b), in which the drag coefficient of model conductors with a large end-plate was about 7% greater than that of conductors without any cover. This shows that the top end-plate is essential. Third, the bottom end-plates were hollowed to reduce their weight and vertical vibrations. This leads to the possibility that airflow may pass through the hollowed regions and act on the conductors. To exclude this possibility, a comparison test was conducted with the bottom end-plate covered with a sticker (Fig. 10). At a low wind speed (U = 10 m/s), the sticker did not exhibit vertical vibrations. The results are presented in Table 3. The difference in the drag coefficient is less than 2%, implying that the hollowing out of the bottom end-plate is acceptable.

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Fig. 9. A model conductor (a) with an end-plate (b) without any cover, and (c) with an end cover at the top.

Fig. 10. Bottom end-plate covered with a sticker. Table 2 Comparison of the drag coefficients of model conductors with different top-end conditions for U = 10 m/s. Model

CD

75◦ -(a)a 75◦ -(b) 75◦ -(c)

1.053 0.934 0.931

a 75◦ -(a) indicates φ = 75◦ and the conductor has an end-plate at the top.

Table 3 Comparison of the drag coefficients of model conductors with different bottom end-plates for U = 10 m/s.

a

Model

CD

60◦ ,a four-bundled conductors without any sticker 60◦ , four-bundled conductors with a sticker

0.581 0.568

60◦ indicates φ = 60◦ .

Finally, after the completion of the major tasks, such as choosing the materials and lengths of conductors, hollowing out the bottom end-plates, and providing a rigid connection between the models and the bottom end-plates, the peak frequencies of the amplitude spectra of the model conductors and setup were obtained by performing the fast Fourier transform of force data. The peak frequencies were greater than 30 Hz and therefore far from the peak frequency of the wind spectrum (Table 4). Furthermore, no vibrations were observed in the model conductors and test setup during a trial run of the HFFB test. This implied that although the top of the conductors was free and the test setup was held up by the strut, the requirements of the HFFB test – small weight and high stiffness – were still met.

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φ φ φ φ

= 45◦ , = 45◦ , = 60◦ , = 75◦ ,

N N N N

=1 = 4, θ = 45◦ = 6, θ = 18◦ = 8, θ = 22.5◦

fx (Hz)

fy (Hz)

fz (Hz)

34.05 33.98 33.98 33.98

34.71 33.98 33.75 33.98

33.05 32.98 31.75 33.25

Fig. 11. Variation of the aerodynamic coefficients of a single conductor with the wind speed (90◦ indicates φ = 90◦ ).

3.2. Effect of Reynolds number The Reynolds number was calculated using the expression Re = UD/ν , where D is the characteristic dimension (conductor diameter d in the present study; see Fig. 2) and ν is the kinematic viscosity. It plays an important role in flow separation, reattachment, and vortex shedding around a circular cylinder. The CD -Re curve of a circular cylinder is typically divided into four parts: subcritical region, critical region, upper transition region, and postcritical region (Demartino and Ricciardelli, 2017). Fig. 11 shows the variation of the aerodynamic coefficients of a single conductor with the wind speed. The CD -Re curve is basically a twofold line, an approximately linear descending part followed by a plateau. The inflection point corresponds to U = 20 m/s and therefore to Re = 4.7 × 104 , irrespective of the wind incidence angle. This value is two orders of magnitude smaller than the Re value of the post-critical region of a circular cylinder (about 3.5 × 106 ), partly because of the effect of the surface roughness (Achenbach, 1971), which is much larger for conductors than a common circular cylinder. With an increase in the wind incidence angle, CD and Cn increase, while CZ decreases. Cp behaves differently, and its largest value corresponds to φ = 75◦ . The following discussion will focus on Cn and Cp at U = 20 m/s, if the wind speed is not mentioned specifically, because these are the most important parameters in practical design.

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Fig. 12. Comparison of test results for Cn,1 (90◦ ).

The drag coefficient of JL/G1A-630/45 at φ = 90◦ , Cn,1 (90◦ ), has been reported in many studies. A comparison of the reported values is presented in Fig. 12. The present test results agree well with the results of Sun and Xie (2013) in the descending part and with the results of Wang et al. (2016) in the plateau part. The inflection point lies between 4 × 104 and 5 × 104 , while the drag coefficients in the plateau region are more scattered, ranging from 0.85–1.03. The model conductor used in the present test was of the same length as that used by Wang et al. (2016); this length was only one-third of the length of the model conductor used by Sun and Xie (2013). Owing to the end effect (Demartino and Ricciardelli, 2017), the drag coefficient in the plateau region of present test is similar to that of Wang et al. (2016) and 10% smaller than that of Xie et al. (2013b). The model conductor used by Zuo et al. (2015) was longer than that used in the present study. However, no strut was used by Zuo et al. (2015) for preventing the influence of the boundary layer created at the bottom of the wind tunnel, and the stiffness of their conductors (which was also installed vertically) was low. Therefore, the drag coefficient obtained by them is the smallest in Fig. 12. On the basis of test results of past studies and the current study, it is recommended that Cn,1 (90◦ ) for JL/G1A-630/45 be considered as 1.0 in practical design. 3.3. Effect of angle of attack As shown in Figs. 13 and 14(a), four-bundled conductors were used to explain the variation of the normal and axial force coefficients with the angle of attack. The conductors are numbered clockwise, and the wind arrives from the right side. It has been previously observed that the shielding zone, or the influence zone of the wake of a conductor, is about ±6d wide in the across-wind direction and more than 30d long in the along-wind direction (Xu et al., 2018). The acrosswind extent is more important than along-wind extent. The shielding effect of four-bundled conductors can be divided 1 2 and ⃝→ 4 3 , where the into two types. One type corresponds to θ = 0◦ and θ = 15◦ , and it can be expressed as ⃝→ ⃝ ⃝ arrow indicates that shielding effect exists and the conductor on the right side is the shielded conductor. The other type 1 3 . Figs. 13(a) and (b) show that the along-wind distance corresponds to θ = 30◦ and θ = 45◦ , and it is expressed as ⃝→ ⃝ 1 and ⃝ 2 changes slowly (0.4d) at θ = 0◦ –15◦ , while the across-wind distance increases by 3d. Therefore, between ⃝ 3 is ‘‘captured’’ by the wake of ⃝ 1 after its ‘‘escape’’ from the in Fig. 14(a), Cn increases. Fig. 13(c) and (d) show that ⃝ 4 . At θ = 30◦ –45◦ , the along-wind distance between ⃝ 3 and ⃝ 1 increases by 0.5d, while the across-wind shielding of ⃝ distance decreases by 4.3d. Therefore, Cn decreases as well. The across-wind distance is larger at θ = 45◦ compared to θ = 0◦ . Furthermore, only one conductor is shielded at θ = 45◦ as against two conductors at θ = 0◦ . Therefore, Cn is larger at θ = 45◦ . The case θ = 22.5◦ is additional. It shows that there is little change in Cn at θ = 15◦ –30◦ . Similar reasoning can be used to explain the curves in Fig. 14(b)–(f). The maximum aerodynamic coefficients of the four-, six-, and eight-bundled conductors can be observed at θ = 30◦ , θ = 20◦ , and θ = 12◦ , respectively, while the minimum values correspond to θ = 0◦ . These value are similar to those obtained by Xie et al. (2013b); in their study, the minimum values were at θ = 0◦ , while the maximum values were at θ = 27◦ , θ = 12◦ , and θ = 13.5◦ , respectively. In Fig. 14, the values of Cn and Cp for different values of φ can be ordered in decreasing order as 90◦ > 75◦ > 60◦ > 45◦ (Cn ) and 75◦ > 45◦ > 60◦ > 90◦ (Cp ), which are the same as the order in Fig. 11(d). Noticeably, at φ = 90◦ , Cp of the eight-bundled conductors is almost zero, while the Cp values of the four- and six-bundled conductors are about 0.1 and 0.04. The cause of this inconsistency was a deviation from the standard test operation: the conductors were not moved to their original position as suggested in Section 2.2. Therefore, the airflow impacted the side faces or even the rear side of the rectangular plate (Fig. 3(d)), producing slight flow separation and local 3D flow. Accordingly, vertical forces increased. It should be noted that the standard test operation was followed in all other respects.

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Fig. 13. Locations of four-bundled conductors at different angles of attack. Table 5 Comparison of the maximum and minimum shielding effect coefficients of the different conductors obtained in the present study and those reported by a previous study for φ = 90◦ . Test

4-bundled min

4-bundled max

6-bundled min

6-bundled max

8-bundled min

8-bundled max

Present test Xie et al. (2013b)

0.81 0.79

0.94 0.95

0.82 0.79

0.93 0.94

0.83 0.80

0.95 0.95

The variation of the angle of attack coefficients for normal and axial forces, WAAC n and WAAC p respectively, with the wind speed is shown in Fig. 15. The WAAC n values of the four-bundled conductors for wind incidence angles of 90◦ , 75◦ , and 60◦ are close to each other, and they decrease from 30%–35% to 15%–20%. WAAC n at φ = 45◦ is the smallest, and it decreases from 23% to 13%. WAAC p of the eight-bundled conductors decreases from 40% to 20% at φ = 75◦ , while at φ = 45◦ , it decreases from 20% to 15%. WAAC p curves for φ = 90◦ and φ = 60◦ are not shown in Fig. 15(b) because the corresponding Cp is too small (less than 0.1), as evident in Fig. 14. For such low values, the error in the calculation of WAAC p is unpredictable (see Eq. (3)). As shown in Fig. 16, the angle of attack coefficient of the normal force initially increased and then decreased with an increase in the wind incidence angle. WAAC n and WAAC p were within 10%–20% and 10%–25%, respectively. The WAAC n values of the four-, six-, and eight-bundled conductors were 17.9%, 16.5%, and 13.6% at φ = 90◦ , respectively. These values are similar to those obtained by Xie et al. (2013b); these authors obtained values of 21.5%, 19.2%, and 15%, respectively. From Figs. 15 and 16, it is apparent that the change in the aerodynamic coefficients resulting from a change in the angle of attack exceeds 10%. 3.4. Shielding effect The variation of the shielding effect coefficient of the normal force, SEC n , with the wind incidence angle and bundle number is shown in Fig. 17. SEC n is related to the angle of attack (see Eq. (4)). The maximum and minimum SEC n values, namely, SEC n,max and SEC n,min , correspond to the angles of attack responsible for the largest and smallest normal force coefficients, respectively (Fig. 14). SEC n,max is within 0.85–0.95 and SEC n,min is within 0.75–0.85, implying that the shielding effect may lead to the overestimation of the wind load by up to 25%. The variation of SEC n with both wind incidence angle and bundle number is less than 10%. It is recommended that the SEC n value of 0.95 be used in practical design. A comparison of the minimum and maximum shielding effect coefficients of the different conductors obtained in the present test and those obtained by Xie et al. (2013b) for φ = 90◦ is presented in Table 5. The difference is less than 4%, indicating the reliability of the present test results. 3.5. Effect of wind incidence angle The variation of the wind incidence angle coefficient of the normal force, WIAC n , with the angle of attack and wind incidence angle is shown in Fig. 18. Auxiliary lines of sin2 φ are drawn for validating the independence principle. The independence principle is applicable to those cases for which WIAC n = sin2 φ . As evident in Fig. 18(a), the variation of

H.J. Xu, H.Z. Deng, X.Y. Hu et al. / Journal of Fluids and Structures 91 (2019) 102702

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Fig. 14. Variation of the normal and axial force coefficients with the angle of attack under skew winds (90◦ indicates φ = 90◦ ).

WIAC n with the angle of attack is less than 2%, implying that WIAC n and the angle of attack (or SEC, which is strongly related to the angle of attack) can be treated as independent parameters. WIAC n at φ = 45◦ is smaller than sin2 φ , the deviation being less than 0.06. WIAC n at φ = 60◦ is larger than sin2 φ (deviation < 0.03), while WIAC n at φ = 75◦ is smaller than sin2 φ (deviation < 0.1). The ratios of the deviations to sin2 φ are 0.12, 0.03, and 0.11 for WIAC n at the φ values of 45◦ , 60◦ , and 75◦ , respectively. These ratios indicate that the normal force on conductors is likely to be overestimated if the independence principle is used. However, the independence principle is still appropriate for practical design.

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Fig. 15. Variation of the angle of attack coefficient with the wind speed (90◦ indicates φ = 90◦ ).

Fig. 16. Variation of the angle of attack coefficient with the wind incidence angle for the different conductors (‘‘4-’’ indicates four-bundled conductors).

Fig. 17. Variation of the shielding effect coefficient with (a) φ and (b) N (4-max denotes the maximum shielding effect coefficient of the four-bundled conductors and 90◦ indicates φ = 90◦ ).

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Fig. 18. Variation of the wind incidence angle coefficient with θ and φ for the different conductors (4–45◦ indicates four-bundled conductors at

φ = 45◦ ).

Fig. 19. Variation of the scale factor with the angle of attack (4–90◦ indicates four-bundled conductors at φ = 90◦ )

In Fig. 19, the variation of the scale factor with the angle of attack is less than 0.03. This implies that SF and the angle of attack, or SEC, can also be treated as independent parameters. The SF values at φ = 60◦ and φ = 90◦ are less than 0.1. Therefore, the corresponding axial wind load can be neglected. It should be noted that the scale factor of the four- and six-bundled conductors at φ = 90◦ is nonzero. The reason for this is identical to that for nonzero Cp under the same test conditions: the nonstandard test operation (see Fig. 14(b) and (d)). At φ = 75◦ and φ = 45◦ , the ranges of SF are 0.2–0.3 and 0.4–0.55, respectively. The axial wind loads in these cases should be considered in practical design. In the Chinese code (DL/T 5154-2012, 2013), the recommended values of SF at φ = 60◦ and φ = 45◦ are 0 and 0.3, respectively. These values are lower than those obtained in the present study. On the basis of the present test results, it is recommended that conservatively, SF values of 0.3 and 0.55 be used for φ values of 75◦ and 45◦ , respectively, and that the SF value be considered to be zero at other φ values. 3.6. Calculation method for normal and axial force coefficients of multi-bundled conductors under skew winds In the discussions in Sections 3.2–3.5, SEC n , WIAC n , and SF were stated to be independent. Therefore, the normal and axial force coefficients of multi-bundled conductors under skew winds can be calculated from the expression

{

Cn = Cn,1 (90◦ ) · SECn,max · WIACn Cp = Cn · SF

The recommended values for the parameters in Eq. (7) are as follows: (i) Cn,1 (90◦ ) = 1.0;

(7)

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(ii) SECn,max = 0.95; (iii) WIACn{= sin2 φ ; 0.55, φ = 45◦ 0.3, φ = 75◦ (iv) SF = 0, other φ 4. Conclusions In this study, inclined model conductors were designed to simulate various wind incidence angles. The aerodynamic coefficients of multi-bundled conductors were measured under skew winds by conducting a wind tunnel test, and the influence of the Reynolds number, the angle of attack, the shielding effect, and the wind incidence angle on them was investigated. The applicability of the independence principle to the calculation of the normal wind load on multi-bundled conductors under skew winds was shown. The conclusions of this study are as follows. (i) The use of the top end-plate and the hollowing out of the bottom end-plate are acceptable in the wind tunnel test. (ii) The CD -Re curve of a single conductor consists of a descending part and a plateau. Re corresponding to the inflection point is 4.7 × 104 , which is two orders of magnitude smaller than that of a circular cylinder. This is partly because of the rougher surface of the conductor. (iii) The aerodynamic coefficients of multi-bundled conductors are strongly dependent on the angle of attack. Their maximum values for the four-, six-, and eight-bundled conductors under skew winds were observed at the θ values of 30◦ , 20◦ , and 12◦ , respectively, and their minimum values were observed at θ = 0◦ . The variation of the aerodynamic coefficients resulting from a change in the angle of attack exceeded 10%. (iv) The shielding effect may lead to the overestimation of the wind load by up to 25%. A shielding effect coefficient of 0.95 is recommended for practical design. (v) The independence principle slightly overestimates the normal wind load. However, it is applicable to multi-bundled conductors under skew winds. (vi) The axial wind load should not be neglected at the φ values of 75◦ and 45◦ . For these φ values, it is recommended that the axial loads be considered as 0.3 and 0.55 times the normal wind load, respectively. (vii) The shielding effect coefficient, wind incidence angle coefficient, and scale factor defined in the present paper can be treated as independent parameters. A simple method for calculating the normal and axial force coefficients (under skew winds) from these parameters is proposed. The present study was conducted by considering specific wind incidence angles and conductor types for uniform flow. Further tests and numerical simulations are necessary for improving the accuracy of the calculation of wind loads on multi-bundled conductors under skew winds. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 51578421), which is gratefully acknowledged. References Achenbach, E., 1971. Influence of surface roughness on the cross-flow around a circular cylinder. J. Fluid Mech. 46 (2), 321–335. 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