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The near wake of a model horizontal-axis wind turbine
Renewable Energy 22 (2001) 461–472 www.elsevier.nl/locate/renene
The near wake of a model horizontal-axis wind turbine Part 3: properties of the tip and hub vortices P.R. Ebert 1, D.H. Wood
*
Department Mechanical Engineering, University of Newcastle, Callaghan, NSW 2308, Australia Received 2 March 2000; accepted 27 April 2000
Abstract This paper concludes a series on the formation and development of the near-wake of a model horizontal-axis wind. The three-dimensional mean velocity and turbulence fields were obtained at six axial locations within two chord lengths of the blades for three operating conditions: stalled flow over the blades, close to optimum performance, and approaching runaway. Here we concentrate on the tip and hub vortices. For the second and third conditions, the hub vortex quickly becomes a sheet of vorticity spread over the cylindrical centrebody. It is suggested that the ‘trailing vorticity’ at the hub is formed by the skewing of the circumferential vorticity in the upstream boundary layer as it encounters the blades, so that the ‘bound’ vorticity of the blades simply passes through the centrebody surface. The tip vortices, which were shown in Part 2 to contain increasing amounts of angular momentum as the tip speed ratio increases, are associated with large variations in all velocity components and high levels of turbulence. The pitch of the helical tip vortices decreases with increasing tip speed ratio. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction This paper is the last of three describing measurements in the near-wake of a twobladed, model horizontal-axis wind turbine for three values of the tip speed ratio, l. Ref. [1] described the experimental arrangements and the measurement techniques, which result in an approximation to the three-dimensional mean flow as seen by an * Corresponding author. 1 Present address: Energy Technology and Environment Branch, Western Power Corporation, 363 Wellington St, Perth, WA 6000, Australia. 0960-1481/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 1 4 8 1 ( 0 1 ) 0 0 0 9 6 - 3
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Nomenclature Cp c p R r t U0 U1 U,V,W ˙ W x
˙ /rU03pR2, turbine power coefficient 2W blade chord (m) vortex pitch tip radius (m) radius (m) time (s) free-stream velocity or wind speed (m/s) average velocity through the blades (m/s) mean velocity components in cylindrical polar co-ordinates power extract from the flow (W) downwind direction from trailing edge
Greek characters d boundary layer thickness ⌫=2prW circulation (m2/s) q circumferential direction location of vortex ‘centre’ qp l tip speed ratio ⍀ vorticity
observer rotating with the blades at a fixed distance downwind of them. It will be assumed that serious readers of this paper are familiar with Refs [1] and [2]. Ref. [1] includes a summary of the small amount of relevant literature, as well as results pertaining to the bound circulation on the blades and the global balance of angular momentum. Ref. [2] shows the general development of the mean velocities, turbulence levels, axial vorticity, and angular momentum, in the form of contour plots for each axial location and l. There is a wealth of information in those plots, which tends to hide the detail of the formation and development of the two most important components of the wake: the tip and hub vortices. These vortices are the subject of this paper. The two, untwisted NACA 4418 blades had a constant chord of 60 mm and a tip radius of 250 mm. The centrebody radius was 55 mm. Hot-wire measurements were taken at six axial locations within two chord lengths of the blades. They comprise all three mean velocities U, V, and W (in the axial or wind, radial, and circumferential directions respectively), and the six Reynolds stresses for three values of l: 2, 4 and 6. From the mean velocities we estimated the three components of mean vorticity using (mainly) central differences and ignoring the gradients in the direction of the wind; thus the axial vorticity, ⍀x, the only vorticity component determined exclusively by cross-wind gradients, is likely to be the most accurate. At the lowest l, the
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blade angles of attack were high and there is evidence of separation in the form of high levels of the turbulent kinetic energy in the blade wakes [2]. The largest power coefficient, Cp, occurred at l=4, where both the bound circulation on the blade and the velocity defect in the wake were almost constant with radius. These closely related features are associated with the simplest of the three wake structures. l=6 had a significantly lower Cp and the bound circulation decreased rapidly towards the tip. At all tip speed ratios, the tip vortex was clearly identifiable by the high levels of turbulence, angular momentum, and axial velocity. Ref. [2] demonstrated that the tip vortices contain increasing amounts of angular momentum as l increases. This implies that runaway, where no power is extracted, occurs when the kinetic energy extracted from the wake is balanced by the work done against the angular momentum flux in the tip vortices. It is essential, therefore, to document the structure of the trailing vortices to assist in developing a consistent and rational computational model capable of predicting blade performance over all the entire operating range. The work described in these papers is an attempt in that direction.
2. Results Figs. 1 and 2 show the downstream development of the axial vorticity, ⍀x and, in the hub region for l=4 and 6 respectively. Only three of the six measurement stations are shown in the figures. The figures were produced by the Tecplot software as ‘flood contours’ from the hot-wire data taken on the data meshes such as that shown in Fig. 3 of Ref. [1]. The hub radius (55 mm) is outlined and the blades are shown in their correct position relative to the flow field, when viewed by an observer rotating (clockwise) with the blades. All the plots display a gap in the data, which gets larger as the angular velocity of the blades was increased to increase l. The gap is a consequence of the need to finish processing the acquired hot-wire data before sampling was restarted by the blade passing the trigger point. Contour levels are given on the right of each figure; all velocities are normalised by the freestream (wind) velocity and all lengths by the turbine radius. The arrows indicate the secondary velocity direction and magnitude. Close to the blades and near the hub, there are regions of vorticity of both signs, which are probably associated with the ‘horseshoe’ vortex that commonly wraps around any large obstacle in a boundary layer. If the obstacle is non-lifting, then the horseshoe vortex must have antisymmetrical regions of equal but opposite vorticity. As in the (stationary) lifting-wing/body measurements of Wood and Westphal [3], the positive and negative regions can be associated with the suction and pressure surface of the blade respectively. However, in the present case, rotation, and the effects of skewing of the mean flow, as discussed in the Appendix A, has significantly altered the vortex structure. For example, the two regions in [3] continued downstream and did not interact, whereas the positive vorticity in Figs. 1 and 2 is quickly enveloped at cancelled by the negative vorticity. In both Figs. 1 and 2, the hub vortex can be approximated by a sheet of circumferential vorticity within two chord lengths
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Fig. 1. Downstream development of the hub vortex, l=4. Contours show axial vorticity with scale at right. Arrows show secondary velocity vectors. (a) x/c=0.167, (b) x/c=0.667, (c) x/c=1.33.
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Fig. 2. Downstream development of the hub vortex, l=6. Contours show axial vorticity with scale at right. Arrows show secondary velocity vectors.(a) x/c=0.167, (b) x/c=0.667, (c) x/c=1.33.
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of the blade. This, in turn, suggests that the hub trailing vortices (downstream of the nacelle) can be represented by a single vortex along the axis of rotation of the blades. It should be mentioned that, if the root end of the blade was not attached to the hub over its entire chord, but finished outside the hub boundary layer, then the hub vortex would, most likely, form in a manner similar to the tip vortex as will be discussed below. To demonstrate that the development of the hub vortex as a sheet is not universal for blades attached directly to the nacelle, Fig. 3 shows the corresponding measurements for the lowest l=2. Here there is no sign of positive vorticity within the hub boundary layer, and, while some of the negative vorticity lies within some form of sheet, a significant region of high negative vorticity remains distinct from that sheet and rotates slowly in the anti-clockwise direction. The behaviour of the contribution of the angular momentum to the output power, lUWr, within the tip vortex was discussed in Ref. [2]. It was shown that the contribution increases with increasing l, to levels that represent a significant reduction in the energy extracted from the wake. Here we document the velocities that give rise to those high levels, and the geometry of the helical tip vortex, mainly in terms of its pitch. From the contours of ⍀x shown in Fig. 3 of Ref. [2], the ‘centre’ of the tip vortices was identified as the location of highest positive axial vorticity. The locus of the angular location of those points is shown in Fig. 4 for all l. The vortex pitch, p, is then given by p=(dqp/dx)−1. Given the uncertainty in locating the vortex centre, due to the relatively small number of measurements within the core, it is reasonable to conclude that p is constant in the near-wake. In his analysis of measurements of propeller tip vortices, Wood [4] showed that the pitch in the near-wake is given approximately by 1+U1 p⬇ 2l
(1)
where U1 is the average velocity through the blades. The present results are generally consistent with this relationship, which, in turn, requires the vortex pitch to decrease with increasing l. For the results in Fig. 4, Eq. (1) gives U1=1.0, 0.78, and 0.80, for l=2, 4, 4.and 6 respectively which can be compared to the measurements of U1 in Fig. 4 of Ref. [1]. In the wakes of propellers, e.g. [4], and hovering rotors, e.g. Leishman et al. [5], the pitch changes abruptly from that given by Eq. (1) when the vortex is behind the blade following the one from which it was shed. For a twobladed turbine this occurs at x=pp, so the present measurements do not extend sufficiently to determine whether a similar change occurs in wind turbine wakes. Fig. 5 shows an enlargement of Fig. 1(b) of Ref. [2] in the vicinity of the tip vortex for l=4, and Fig. 6 shows the distribution of the velocity components along the radial and circumferential lines shown in Fig. 5. There are high levels of the axial velocity, U, within the vortex, but as demonstrated in [2], U is remarkably uniform outside the vortex, where the radial velocity V is nearly zero. Outside the hub vortex, which ‘induces’ it, the circumferential velocity W, decreases approximately as r⫺1. The shape of the tip vortex in Fig. 5 is generally consistent with the expectation that the measurements represent a slice through nearly circular vortices at an incli-
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Fig. 3. Downstream development of the hub vortex, l=2. Contours show axial vorticity with scale at right. Arrows show secondary velocity vectors.(a) x/c=0.167, (b) x/c=0.667, (c) x/c=1.33.
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Fig. 4.
Downstream location of the tip vortices for values of l given in the figure.
Fig. 5. Detail of tip vortex, l=4, x/c=1.667. Contours show axial vorticity with scale at right. Arrows show secondary velocity vectors. The dashed radial and circumferential lines intersect at the vortex centre.
nation that increases with increasing l. However, Fig. 7 shows that the approximately oval vortex shown in Fig. 5 does not transform into a circular vortex if the coordinate system is rotated by the angle given by the vortex pitch. It appears that significantly more measurements at an improved spatial resolution are required to fully delineate the behaviour of the tip vortices.
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Fig. 6. Axial, radial, and circumferential velocity distributions through the vortex in Fig. 5. Figures on the left show the radial distributions along the lines of the same type in Fig. 5. Figures on the right show circumferential distributions along the lines of the same type. The origin for q is arbitrary.
3. Conclusions This paper has concentrated on those features of the three-dimensional flowfield documented in Ref. [2], that pertain to the hub and the tip vortices. In comparison to propeller and hovering rotor wakes, there is a dearth of information on the formation and development of the vortex structure in the wake of wind turbines. It was shown that the tip vortex pitch decreased with increasing tip speed ratio, and that it is characterised by large values and large gradients in the mean axial and circumferential velocities. Unfortunately, too few measurements were taken within the tip vortex to adequately delineate its structure, but there is no doubt that the tip vortex contains increasing amounts of angular momentum that drain the power output as the tip speed ratio increases. The hub vortex quickly formed a vortex sheet at the two highest tip speed ratios in a manner that differs significantly from the formation of the trailing vorticity from a lifting-wing/body junction. It is shown in the Appendix that this process is consistent with the necessary change in direction of the mean flow as it passes the blades (and hence generation of circulation). In other words, the downstream axial vorticity could well have been formed from the circumferential vorticity in the boundary layer upstream of the blades. Other blade attachment geometries may cause the hub trailing vorticity to be formed in a different manner.
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Fig. 7. Transformation of vorticity contours in Fig. 5 to local co-ordinates. ⍀a is the vorticity along the vortex axis, y* and z* are the local approximation to the radial and circumferential directions respectively in the plane normal to the axis.
Acknowledgements This work was supported by the Australian Research Council. During the experimental work PRE held an Australian Postgraduate Research Award; we thank the Western Power Corporation for allowing him computer time to work on preparing this paper. Mr J.J. Smith wrote most of the data acquisition software. Technical assistance was provided by Msrs J.D. Walton, I. Miller, and R. Reece. The efforts of the Mechanical Engineering Department’s workshop staff in building the model turbine and dynamometer are also acknowledged.
Appendix A. The production of hub axial vorticity from the circumferential vorticity in the Upstream Boundary Layer This appendix uses a very approximate treatment of the Squire-Winter-Hawthorne (SWH) secondary flow formula to demonstrate that the hub vorticity behind a hori-
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zontal-axis wind turbine can be formed entirely by the skewing of the circumferential vorticity in the boundary layer upstream of the blades. This skewing is related to the change in the mean flow direction as a consequence of the extraction of angular momentum, and hence power, from the flow. In an isolated wing-body junction, there is no net change in direction [3]. In the ideal case (and to a good approximation in the present experiment) there is no swirl (circumferential motion or axial vorticity, ⍀x) in the airflow upstream of the blades (Taylor [6]). Downwind of the blades, the swirl caused by the hub vortex is associated with the angular momentum and power extracted by the blades. We hypothesise that the hub vortex can be produced entirely by the skewing of the mean streamlines near the hub as they encounter the blades rather than shedding of the bound vorticity as occurs at the tip. If the hypothesis is correct, ⍀x is produced from ⍀q. and according to the SWH formula:
冉 冊 冉冊
d ⍀x d W ⫽ dx ⍀q dx U
(2)
e.g. Bradshaw [7]. As both W and ⍀x and are zero upstream of the blades, we can use Eq. (2) to estimate the magnitude of ⍀x behind the blades as ⍀x W ⬇ ⍀q U in the downstream flow. Now ⍀x⬇W/d where W is the circumferential velocity at the boundary layer edge, and d is the layer thickness. If this approximate equation is ‘integrated’ over the hub boundary layer we have 2prhd⍀x⬇⌫⬇2prhW
(3)
on the further assumption that the hub radius rhÀd. Eq. (3) suggests that the skewing of the circumferential vorticity in the upstream boundary layer as it passes the blades is sufficient to generate the downstream circulation associated with axial vorticity. Presumably the high local solidity of the hub region allows that skewing to be shared equally by all parts of the boundary layer, giving rise to the vortex sheet described in the main text.
References [1] Ebert PR, Wood DH. The near wake of a model horizontal-axis wind turbine. Part 1: experimental Arrangements and Initial results. Ren Energy 1997;12:225–43. [2] Ebert PR, Wood DH. The near wake of a model horizontal-axis wind turbine. Part 2: general features of the three-dimensional flowfield. Ren Energy 1999;18:513–34. [3] Wood DH, Westphal RV. Measurements of the flow around a lifting-wing/body junction. AIAA J 1992;30:6–12. [4] Wood DH. Simple equations for helical vortex wakes. J Aircraft 1992;30:6–12. [5] Leishman JG, Baker A, Coyne A. Measurements of rotor tip vortices using three-component laser Doppler anemometry. J Am Helicopter Soc 1996;41:342–53.
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[6] Taylor GI. The ‘rotational’ inflow factor in propeller theory. In: Batchelor GK, editor. Scientific Papers III. Cambridge: Cambridge University Press, 1963:59–65. [7] Bradshaw P. Turbulent secondary flow. Annu Rev Fluid Mech 1987;19:53–74.
The near wake of a model horizontal-axis wind turbine Part 3: properties of the tip and hub vortices P.R. Ebert 1, D.H. Wood
*
Department Mechanical Engineering, University of Newcastle, Callaghan, NSW 2308, Australia Received 2 March 2000; accepted 27 April 2000
Abstract This paper concludes a series on the formation and development of the near-wake of a model horizontal-axis wind. The three-dimensional mean velocity and turbulence fields were obtained at six axial locations within two chord lengths of the blades for three operating conditions: stalled flow over the blades, close to optimum performance, and approaching runaway. Here we concentrate on the tip and hub vortices. For the second and third conditions, the hub vortex quickly becomes a sheet of vorticity spread over the cylindrical centrebody. It is suggested that the ‘trailing vorticity’ at the hub is formed by the skewing of the circumferential vorticity in the upstream boundary layer as it encounters the blades, so that the ‘bound’ vorticity of the blades simply passes through the centrebody surface. The tip vortices, which were shown in Part 2 to contain increasing amounts of angular momentum as the tip speed ratio increases, are associated with large variations in all velocity components and high levels of turbulence. The pitch of the helical tip vortices decreases with increasing tip speed ratio. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction This paper is the last of three describing measurements in the near-wake of a twobladed, model horizontal-axis wind turbine for three values of the tip speed ratio, l. Ref. [1] described the experimental arrangements and the measurement techniques, which result in an approximation to the three-dimensional mean flow as seen by an * Corresponding author. 1 Present address: Energy Technology and Environment Branch, Western Power Corporation, 363 Wellington St, Perth, WA 6000, Australia. 0960-1481/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 1 4 8 1 ( 0 1 ) 0 0 0 9 6 - 3
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Nomenclature Cp c p R r t U0 U1 U,V,W ˙ W x
˙ /rU03pR2, turbine power coefficient 2W blade chord (m) vortex pitch tip radius (m) radius (m) time (s) free-stream velocity or wind speed (m/s) average velocity through the blades (m/s) mean velocity components in cylindrical polar co-ordinates power extract from the flow (W) downwind direction from trailing edge
Greek characters d boundary layer thickness ⌫=2prW circulation (m2/s) q circumferential direction location of vortex ‘centre’ qp l tip speed ratio ⍀ vorticity
observer rotating with the blades at a fixed distance downwind of them. It will be assumed that serious readers of this paper are familiar with Refs [1] and [2]. Ref. [1] includes a summary of the small amount of relevant literature, as well as results pertaining to the bound circulation on the blades and the global balance of angular momentum. Ref. [2] shows the general development of the mean velocities, turbulence levels, axial vorticity, and angular momentum, in the form of contour plots for each axial location and l. There is a wealth of information in those plots, which tends to hide the detail of the formation and development of the two most important components of the wake: the tip and hub vortices. These vortices are the subject of this paper. The two, untwisted NACA 4418 blades had a constant chord of 60 mm and a tip radius of 250 mm. The centrebody radius was 55 mm. Hot-wire measurements were taken at six axial locations within two chord lengths of the blades. They comprise all three mean velocities U, V, and W (in the axial or wind, radial, and circumferential directions respectively), and the six Reynolds stresses for three values of l: 2, 4 and 6. From the mean velocities we estimated the three components of mean vorticity using (mainly) central differences and ignoring the gradients in the direction of the wind; thus the axial vorticity, ⍀x, the only vorticity component determined exclusively by cross-wind gradients, is likely to be the most accurate. At the lowest l, the
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blade angles of attack were high and there is evidence of separation in the form of high levels of the turbulent kinetic energy in the blade wakes [2]. The largest power coefficient, Cp, occurred at l=4, where both the bound circulation on the blade and the velocity defect in the wake were almost constant with radius. These closely related features are associated with the simplest of the three wake structures. l=6 had a significantly lower Cp and the bound circulation decreased rapidly towards the tip. At all tip speed ratios, the tip vortex was clearly identifiable by the high levels of turbulence, angular momentum, and axial velocity. Ref. [2] demonstrated that the tip vortices contain increasing amounts of angular momentum as l increases. This implies that runaway, where no power is extracted, occurs when the kinetic energy extracted from the wake is balanced by the work done against the angular momentum flux in the tip vortices. It is essential, therefore, to document the structure of the trailing vortices to assist in developing a consistent and rational computational model capable of predicting blade performance over all the entire operating range. The work described in these papers is an attempt in that direction.
2. Results Figs. 1 and 2 show the downstream development of the axial vorticity, ⍀x and, in the hub region for l=4 and 6 respectively. Only three of the six measurement stations are shown in the figures. The figures were produced by the Tecplot software as ‘flood contours’ from the hot-wire data taken on the data meshes such as that shown in Fig. 3 of Ref. [1]. The hub radius (55 mm) is outlined and the blades are shown in their correct position relative to the flow field, when viewed by an observer rotating (clockwise) with the blades. All the plots display a gap in the data, which gets larger as the angular velocity of the blades was increased to increase l. The gap is a consequence of the need to finish processing the acquired hot-wire data before sampling was restarted by the blade passing the trigger point. Contour levels are given on the right of each figure; all velocities are normalised by the freestream (wind) velocity and all lengths by the turbine radius. The arrows indicate the secondary velocity direction and magnitude. Close to the blades and near the hub, there are regions of vorticity of both signs, which are probably associated with the ‘horseshoe’ vortex that commonly wraps around any large obstacle in a boundary layer. If the obstacle is non-lifting, then the horseshoe vortex must have antisymmetrical regions of equal but opposite vorticity. As in the (stationary) lifting-wing/body measurements of Wood and Westphal [3], the positive and negative regions can be associated with the suction and pressure surface of the blade respectively. However, in the present case, rotation, and the effects of skewing of the mean flow, as discussed in the Appendix A, has significantly altered the vortex structure. For example, the two regions in [3] continued downstream and did not interact, whereas the positive vorticity in Figs. 1 and 2 is quickly enveloped at cancelled by the negative vorticity. In both Figs. 1 and 2, the hub vortex can be approximated by a sheet of circumferential vorticity within two chord lengths
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Fig. 1. Downstream development of the hub vortex, l=4. Contours show axial vorticity with scale at right. Arrows show secondary velocity vectors. (a) x/c=0.167, (b) x/c=0.667, (c) x/c=1.33.
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Fig. 2. Downstream development of the hub vortex, l=6. Contours show axial vorticity with scale at right. Arrows show secondary velocity vectors.(a) x/c=0.167, (b) x/c=0.667, (c) x/c=1.33.
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of the blade. This, in turn, suggests that the hub trailing vortices (downstream of the nacelle) can be represented by a single vortex along the axis of rotation of the blades. It should be mentioned that, if the root end of the blade was not attached to the hub over its entire chord, but finished outside the hub boundary layer, then the hub vortex would, most likely, form in a manner similar to the tip vortex as will be discussed below. To demonstrate that the development of the hub vortex as a sheet is not universal for blades attached directly to the nacelle, Fig. 3 shows the corresponding measurements for the lowest l=2. Here there is no sign of positive vorticity within the hub boundary layer, and, while some of the negative vorticity lies within some form of sheet, a significant region of high negative vorticity remains distinct from that sheet and rotates slowly in the anti-clockwise direction. The behaviour of the contribution of the angular momentum to the output power, lUWr, within the tip vortex was discussed in Ref. [2]. It was shown that the contribution increases with increasing l, to levels that represent a significant reduction in the energy extracted from the wake. Here we document the velocities that give rise to those high levels, and the geometry of the helical tip vortex, mainly in terms of its pitch. From the contours of ⍀x shown in Fig. 3 of Ref. [2], the ‘centre’ of the tip vortices was identified as the location of highest positive axial vorticity. The locus of the angular location of those points is shown in Fig. 4 for all l. The vortex pitch, p, is then given by p=(dqp/dx)−1. Given the uncertainty in locating the vortex centre, due to the relatively small number of measurements within the core, it is reasonable to conclude that p is constant in the near-wake. In his analysis of measurements of propeller tip vortices, Wood [4] showed that the pitch in the near-wake is given approximately by 1+U1 p⬇ 2l
(1)
where U1 is the average velocity through the blades. The present results are generally consistent with this relationship, which, in turn, requires the vortex pitch to decrease with increasing l. For the results in Fig. 4, Eq. (1) gives U1=1.0, 0.78, and 0.80, for l=2, 4, 4.and 6 respectively which can be compared to the measurements of U1 in Fig. 4 of Ref. [1]. In the wakes of propellers, e.g. [4], and hovering rotors, e.g. Leishman et al. [5], the pitch changes abruptly from that given by Eq. (1) when the vortex is behind the blade following the one from which it was shed. For a twobladed turbine this occurs at x=pp, so the present measurements do not extend sufficiently to determine whether a similar change occurs in wind turbine wakes. Fig. 5 shows an enlargement of Fig. 1(b) of Ref. [2] in the vicinity of the tip vortex for l=4, and Fig. 6 shows the distribution of the velocity components along the radial and circumferential lines shown in Fig. 5. There are high levels of the axial velocity, U, within the vortex, but as demonstrated in [2], U is remarkably uniform outside the vortex, where the radial velocity V is nearly zero. Outside the hub vortex, which ‘induces’ it, the circumferential velocity W, decreases approximately as r⫺1. The shape of the tip vortex in Fig. 5 is generally consistent with the expectation that the measurements represent a slice through nearly circular vortices at an incli-
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Fig. 3. Downstream development of the hub vortex, l=2. Contours show axial vorticity with scale at right. Arrows show secondary velocity vectors.(a) x/c=0.167, (b) x/c=0.667, (c) x/c=1.33.
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Fig. 4.
Downstream location of the tip vortices for values of l given in the figure.
Fig. 5. Detail of tip vortex, l=4, x/c=1.667. Contours show axial vorticity with scale at right. Arrows show secondary velocity vectors. The dashed radial and circumferential lines intersect at the vortex centre.
nation that increases with increasing l. However, Fig. 7 shows that the approximately oval vortex shown in Fig. 5 does not transform into a circular vortex if the coordinate system is rotated by the angle given by the vortex pitch. It appears that significantly more measurements at an improved spatial resolution are required to fully delineate the behaviour of the tip vortices.
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Fig. 6. Axial, radial, and circumferential velocity distributions through the vortex in Fig. 5. Figures on the left show the radial distributions along the lines of the same type in Fig. 5. Figures on the right show circumferential distributions along the lines of the same type. The origin for q is arbitrary.
3. Conclusions This paper has concentrated on those features of the three-dimensional flowfield documented in Ref. [2], that pertain to the hub and the tip vortices. In comparison to propeller and hovering rotor wakes, there is a dearth of information on the formation and development of the vortex structure in the wake of wind turbines. It was shown that the tip vortex pitch decreased with increasing tip speed ratio, and that it is characterised by large values and large gradients in the mean axial and circumferential velocities. Unfortunately, too few measurements were taken within the tip vortex to adequately delineate its structure, but there is no doubt that the tip vortex contains increasing amounts of angular momentum that drain the power output as the tip speed ratio increases. The hub vortex quickly formed a vortex sheet at the two highest tip speed ratios in a manner that differs significantly from the formation of the trailing vorticity from a lifting-wing/body junction. It is shown in the Appendix that this process is consistent with the necessary change in direction of the mean flow as it passes the blades (and hence generation of circulation). In other words, the downstream axial vorticity could well have been formed from the circumferential vorticity in the boundary layer upstream of the blades. Other blade attachment geometries may cause the hub trailing vorticity to be formed in a different manner.
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Fig. 7. Transformation of vorticity contours in Fig. 5 to local co-ordinates. ⍀a is the vorticity along the vortex axis, y* and z* are the local approximation to the radial and circumferential directions respectively in the plane normal to the axis.
Acknowledgements This work was supported by the Australian Research Council. During the experimental work PRE held an Australian Postgraduate Research Award; we thank the Western Power Corporation for allowing him computer time to work on preparing this paper. Mr J.J. Smith wrote most of the data acquisition software. Technical assistance was provided by Msrs J.D. Walton, I. Miller, and R. Reece. The efforts of the Mechanical Engineering Department’s workshop staff in building the model turbine and dynamometer are also acknowledged.
Appendix A. The production of hub axial vorticity from the circumferential vorticity in the Upstream Boundary Layer This appendix uses a very approximate treatment of the Squire-Winter-Hawthorne (SWH) secondary flow formula to demonstrate that the hub vorticity behind a hori-
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zontal-axis wind turbine can be formed entirely by the skewing of the circumferential vorticity in the boundary layer upstream of the blades. This skewing is related to the change in the mean flow direction as a consequence of the extraction of angular momentum, and hence power, from the flow. In an isolated wing-body junction, there is no net change in direction [3]. In the ideal case (and to a good approximation in the present experiment) there is no swirl (circumferential motion or axial vorticity, ⍀x) in the airflow upstream of the blades (Taylor [6]). Downwind of the blades, the swirl caused by the hub vortex is associated with the angular momentum and power extracted by the blades. We hypothesise that the hub vortex can be produced entirely by the skewing of the mean streamlines near the hub as they encounter the blades rather than shedding of the bound vorticity as occurs at the tip. If the hypothesis is correct, ⍀x is produced from ⍀q. and according to the SWH formula:
冉 冊 冉冊
d ⍀x d W ⫽ dx ⍀q dx U
(2)
e.g. Bradshaw [7]. As both W and ⍀x and are zero upstream of the blades, we can use Eq. (2) to estimate the magnitude of ⍀x behind the blades as ⍀x W ⬇ ⍀q U in the downstream flow. Now ⍀x⬇W/d where W is the circumferential velocity at the boundary layer edge, and d is the layer thickness. If this approximate equation is ‘integrated’ over the hub boundary layer we have 2prhd⍀x⬇⌫⬇2prhW
(3)
on the further assumption that the hub radius rhÀd. Eq. (3) suggests that the skewing of the circumferential vorticity in the upstream boundary layer as it passes the blades is sufficient to generate the downstream circulation associated with axial vorticity. Presumably the high local solidity of the hub region allows that skewing to be shared equally by all parts of the boundary layer, giving rise to the vortex sheet described in the main text.
References [1] Ebert PR, Wood DH. The near wake of a model horizontal-axis wind turbine. Part 1: experimental Arrangements and Initial results. Ren Energy 1997;12:225–43. [2] Ebert PR, Wood DH. The near wake of a model horizontal-axis wind turbine. Part 2: general features of the three-dimensional flowfield. Ren Energy 1999;18:513–34. [3] Wood DH, Westphal RV. Measurements of the flow around a lifting-wing/body junction. AIAA J 1992;30:6–12. [4] Wood DH. Simple equations for helical vortex wakes. J Aircraft 1992;30:6–12. [5] Leishman JG, Baker A, Coyne A. Measurements of rotor tip vortices using three-component laser Doppler anemometry. J Am Helicopter Soc 1996;41:342–53.
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P.R. Ebert, D.H. Wood / Renewable Energy 22 (2001) 461–472
[6] Taylor GI. The ‘rotational’ inflow factor in propeller theory. In: Batchelor GK, editor. Scientific Papers III. Cambridge: Cambridge University Press, 1963:59–65. [7] Bradshaw P. Turbulent secondary flow. Annu Rev Fluid Mech 1987;19:53–74.