fdwt200801

FLODESIGN, INC. WILBRAHAM, MA.01095 REPORT No. FD 200801 A New Analytical Model for Wind Turbine Wakes by Michael, J. Werle, PhD June 1, 2008 © FloDesign Inc. Wilbraham MA 2 MA 01095 A new. The current model clearly shows that the fundamental root cause of this loss is the deep velocity deficit trough aft of the lead turbine that is only slowly overcome.A New Analytical Model for Wind Turbine Wakes M. Wilbraham MA . see Eq. 380 Main Street. (3a) CT Thrust coefficient. It is first-principles based and overcomes past formulation limitations. 2 and 3 respectively in Fig.) an exact model of the essentially inviscid near wake flow region 2. provides a new pathway for addressing the critical issue of wind farm productivity prediction and design layout. 1 1 Chief Scientist. AIAA Fellow © FloDesign Inc. 3 v properties in the viscous or far wake 0 properties at far wake virtual origin 1. 3 i inviscid or near wake region o downstream outlet conditions see Fig.) a far wake model based on the classical Prandtl/Swain axisymmetric wake analysis. and 3. see Fig. see Fig. (1) ρ fluid density Subscripts a ambient free stream conditions d conditions on down-wind side of wind turbine. 3 u conditions on up-wind side of wind turbine. The model also provides the basis for extending its application to downstream rows of turbines imbedded in a wind farm. PhD1 FloDesign Inc.) a methodology for estimating the length of the intermediate wake based on Prandtl’s turbulent shear layer mixing solution.3 properties at wind turbines 1. verified through comparison with a large compendium of measured results. The model’s predictions reinforce results of recent studies of horizontal axis wind turbine farms indicating that longitudinal turbine spacing of less than 10 diameters lead to severe wake-induced productivity losses. (1) CP Power coefficient. The model. (3b) K Absolute mixing-length related constant for the far wake region Km Relative mixing-length related constant for the intermediate wake region L Nondiensional longitudinal distance between wind turbines. see Fig. 1 p Pressure P Power T Thrust u nondimensional V/Va V Velocity W Nondimensional lateral spacing between wind farm rows. see Eq. 1 x Distance down-wind of a wind turbine X Nondimensional axial distance from a wind turbine. Wilbraham. Wilbraham. Nomenclature A Flow cross sectional area c Circulation related constant of Eq. MA 01095.2. It contains three critical elements: 1. Werle. analytical model is put forward for estimating the impact of a wind turbine on its immediate downstream neighbor as occurs in virtually all existing wind farms. J. see Fig. see Eq. closed form. The recent work of European teams and individual researchers (see Refs. 1. It assumes large enough lateral spacing. 1.. Introduction A large majority of the horizontal axis wind turbines (HAWTs) in use today are employed in wind farm arrays. 1-22). © FloDesign Inc. remains constant across the turbine and thereafter decreases further in the near wake as the pressure rises downstream to pa. W. • The velocity inside the capture streamtube decreases as it approaches the turbine. Turbines 2. like all axisymmetric bodies. The Near Wake: The principle features of the near wake region of Fig.. the near wake. which. then drops abruptly across the turbine and thereafter increases steady in the near wake region to the free-stream value. The fluid dynamic situation is complex not only analytically due to the merging of multiple wakes as one proceeds down a row of turbines (see Fig. 1) as it mixes out with distance and therefore sets up the velocity field approaching a down-wind turbine such as Turbine 2 of Fig. 3 and beyond of Fig. In particular. 2 relevant to current purposes are: • The pressure rises ahead of the turbine as the capture streamtube expands to the blade’s diameter. pa.e. 1 will sometimes deliver about 65% of the Turbine 1 power level. 2. 1-22) with a comprehensive review of the state-of-the-art as of 2003 provided by in Ref. Sorensen and Crespo. The current effort focuses entirely on the first-turbine wake problem. It has long been known that turbines’ individual longitudinal and lateral displacements can strongly influence the wind farm’s power generating efficiency. It has also highlighted the fact that a successful basis for establishing a reliable analytical model for this has not yet emerged. the capture streamtube) grows to a fixed diameter as the pressure approaches pa and the velocity decreases to satisfy mass and momentum conservation. i. in axial length. such as that depicted schematically in Fig. Numerous attempts have been made to provide an accurate model for the velocities in wind turbines wakes (see Refs. has three distinct regimes as depicted in Fig.. a first principals basedEncouraginr comparisons of the wake centerline velocities. second-turbine power degradation and wake growth rate are shown for a very wide collection of data including full scale wind turbines in both off shore or on-shore placements. Placing the turbines too close together reduces the output of all downwind turbines due to the wake velocity deficits of upwind turbines while placing them too far apart reduces the number of turbines and thus the amount of energy extractable from a given property size. The Intermediate Wake: The principle features of the intermediate wake region are: • The pressure is constant and equal to pa. 1-6 for example) show that even with the relatively large longitudinal and lateral separation distances of seven HAWT diameters. • The width of the near wake aft of the turbine (i. to date. Recent studies (see Refs. such that Rows 2 and 3 induce negligible effects on Turbines 1 & 2 of Row 1. Each will be discussed briefly below in order to set the stage for the analytical models developed in the three sections that follow. as well as small scale wind turbine rotors and porous plate simulators tested in wind tunnels. 2. there still exists uncertainty as to how to model the velocity deficit produced in the wake of an individual wind turbine (e. In the following sections. the intermediate wake and the far wake. Turbine 1 of Fig. it is well documented and understood that a wind turbine wake. 1) but it is also difficult to obtain definitive.1. Therefore it is critically important to determine the optimal array configuration in order to maximize a wind farm’s productivity. Wilbraham MA 2 .g. Dp. Both of these effects have direct negative influence on the cost of the power delivered. 11 by Vemeer. The Flow Structure in a Wind Turbine Wake As discussed in most reference texts and numerous papers (see Refs. • This region is dominated by inviscid processes and is known to be of the order of one prop diameter. reliable data for such large structures in open wind settings. modeling the influence of Turbine 1 on Turbine 2. 1-6) must be noted in this regard as it has begun to provide needed insights relative to the flow structure attendant to wind turbines in wind farms. This is the core building block for wind farm modeling.e. has not been satisfactorily attained. ) the Prandtl/Swain far wake analytical model’s applicability is verified through comparisons with a very large compendium of wind turbine related measurements. 2. anywhere within the capture streamtube as: u = 1 + c ⎢1 + i ⎡ ⎢ ⎣ ⎤ ⎥. 2⎥ 1 + 4X ⎦ 2X (1a) where c is related to the total circulation induced by the wind turbine and © FloDesign Inc. 2. Prandtl and Swain (Refs. current models have not employed the thrust coefficient scaling (momentum deficit) dictated by the Prandtl/Swain exact solutions. pp 96-98) provides the exact solution of the Biot Savart Law applied to the spiraling vorticity field aft of a lifting prop to write the nondimensional inviscid centerline velocity. plus an arbitrary constant representing the virtual origin of the far wake which must be determined from the near and intermediate wake behavior. but have focused principally on a far wake like structure. ui. The Near Wake Region As depicted in Fig. Va. power-law scaling for the entire wake in an attempt to improve agreement with measured data. even within the far wake region. 27. due to turbulent mixing. The Far Wake: The principle features of the far wake region are: • The pressure remains constant and equal to pa. • The Prandtl/Swain solution contains two empirical constants: an “absolute” constant applicable to all axisymmetric wakes which must be determined from experiments. none have led to satisfactory results. simple.• • • The centerline velocity remains constant as turbulent mixing increase at the wake outer boundary due to the large radial gradient in the axial velocity. Very encouraging comparisons are provided with measured centerline velocity decay. exact closed form solution of the vorticity equations that was originally developed for propeller-based propulsion. The length of this region is reported to be several diameters long and ends when the mixing layer reaches the centerline and initiates a change in the centerline velocity. 23 or see Refs. second turbine power degradation and lateral wake growth rate for a wide range of thrust coefficients. As a result many have turned to a single.) an algebraically simple composite solution for the entire wake can be constructed using Prandtl’s self-similar solution for a lateral velocity discontinuity to couple the near and far wake solutions. Wilbraham MA 3 . 3. 24 & 27) provided first and second order accurate closed form. 3 for a typical HAWT. the near wake will be considered to be governed by inviscid pressure forces with the turbulent mixing layer that initiates at the turbine having negligible effect to first order on the flow structure. empirically based. 1-6 for example) have not employed the details of the three phase structure of Fig. The implications of the model’s prediction for wind farm spacing and productivity levels are also discussed. The resulting models have generally shown either limited or unsatisfactory agreement with measured data. However. In the following sections. most published attempts to analytically model wind turbine wakes (see Refs. powerlaw solutions for this region using self-similarity concepts that scale with the momentum deficit induced by the wind turbine. The initial lateral spread rate of the mixing layer can be approximated using Prandtl’s self-similar solution (Ref. to date. McCormick (Ref.) the near wake centerline velocity decay can be calculated from a well documented. these limitations will be relieved and it will be shown that: 1. To date. • The lateral spread rate of the mixing region was predicted by Prandtl and Swain to be governed by fractional powers of axial distance and the thrust coefficient of the wind turbine. and 3. • The centerline velocity now begins a steady increase toward the free stream value. 24 & 25 for example) for the turbulent mixing of a planar velocity discontinuity. However. uO of Fig. c can be determined by matching the predicted outlet velocity at downstream infinity. 1 + 4X 2 ⎥ ⎦ 2X (2a) With this. 3. CT=1. For CT < 8/9 (Max Power level) they effectively reach their limit values at X=2 while for CT > 8/9. o 2 2 1 ρA V 1 ρA V 2 2 p a p a ( ) (3b) As indicated in Section 2 above. 2. uO. (3a)-(3b) can be inverted to write that: u o = 1−CT . CT. 1 on the power degradation from Turbine 1 to 2. the far wake scales with the Thrust Coefficient.u 2 o 1+ ⎢ u = 1− i ⎡ ⎢ ⎣ ⎤ ⎥.. o o 2 1 ρA V3 2 p a Power Coefficient: ( )( ) (3a) Thrust Coefficient: A (p − p ) p u d 2 CT ≡ = = 1− u . the Betz Limit of CP =16/27. one can write the power ratio as: © FloDesign Inc. Two important observations can be made here that will be of value in what follows: 1.X ≡ x/D p . (3a). Wilbraham MA 4 . Using the definition given in Eq. (4a) and C P = 1 CT 1 + 1 − CT . Thus Eqs. The case of CT = 8/9 corresponds to attainment of the maximum power output. and wake growth rate. the outlet velocity. especially for CT < 8/9. (1b) For current purposes. both the centerline velocity and capture streamtube take longer to attain their asymptotic levels. 2 [ ] (4b) Figures 4(a) and 4(b) provide the resulting values of the centerline velocity.20-22. The centerline velocity results of Fig. Downstream of the HAWT. 20-22 or 28. D (2b) From any of Refs. Di of the capture streamtube can calculated from mass conservations as: D = i p 1+ u 2u i o . the centerline velocity and capture streamtube very clearly attain their asymptotic levels within one to two diameters regardless of the thrust level. X increase to four or more as stall is approached.e. can be determined from either of the following relations: CP ≡ P 1 ρA V3 2 p a T = u A (p − p ) 1 p p u d 2 = 1+ u 1− u . the width. Di/Dp for a complete range of CT up to the stall limit. i. to that predicted through use of a momentum balance and actuator disk theory (see Refs. ui. which will be taken here as the controlling parameter for the entire problem. Upstream of the HAWT. or 28 for example) to write that 1. 4(a) can be used to begin to assess the influence of the turbine longitudinal spacing of Fig. 26). ui. To that end.e. 2/3 (7a) and the attendant viscous induced centerline velocity. it will be assumed here the HAWT employed has a somewhat idealized thrust and power curves as depicted in Fig. CT1 of Eq. performed in collaboration with Prandtl. Relief of this constraint can be applied straightforwardly for any particular HAWT application of interest. 4. Wilbraham MA 5 . i. as: ⎛ 1/2 ⎞ ⎜ C /X ⎟ T ⎠ uv = 1 − ⎝ 2 2K . (5) & (6) for the full range of Thrust Coefficients and longitudinal spacing up to L = 5. Swain’s work (Ref. so long as the turbulent mixing length hypothesis is valid. for current purposes: 1− u 2 o ⎢1 + ⎡ ⎢ ⎣ V V = u = 1+ 2 a i2 ⎤ ⎥ 2⎥ 1 + 4(L . use will be made of the first order approximation to write the viscous wake growth rate. 5. (2a) at a value of X < L to account for the upstream propagation effect discussed in item 1 directly above. as shown in Fig 5. Thus. With this. (5) is merely Turbine 1’s wake velocity. the levels become extremely low as the Max Power situation is approached and are seen to continue towards zero as the stall level is approached. be determined from the performance characteristics of the particular HAWT being employed. © FloDesign Inc. is known but the corresponding value for Turbine 2. The ratio V2/Va of Eq. as given by Swain as: D D =KC X p T v ( )1 / 3 .P 2 ⎡ ⎤ 3 ⎛ V2 ⎞ CT2 ⎢1 + 1 − CT2 ⎥ ⎣ ⎦ ⎜ ⎟ = ⎜V ⎟ P ⎡ ⎤ 1 ⎝ a ⎠ C ⎢1 + 1 − C ⎥ T1 ⎣ T1 ⎦ (5) where Va is the upstream ambient wind speed approaching Turbine 1 and V2 is the effective wind speed approaching Turbine 2. in turn. the constant K is related to Prandtl’s turbulent mixing length and was designated by Swain as a “universal” constant. The Far Wake Region It is useful to next give attention to modeling of the far wake region. will also establish the pathway by which the three regions become analytically linked for ultimate closure. CT2 . one that must be determined from experiments and will be the same for all axisymmetric bodies generating a momentum deficit in a uniform stream.1) ⎦ 2(L . which. it is reasonable to assume that the thrust coefficients are approximately equal in Eq.1) (6) The Turbine 1 Thrust Coefficient. Dv. calculated by evaluating Eq. 23) to include higher order terms in the asymptotic analysis. a relatively flat thrust curve up to the turbines rated power level. i. must. extended his earlier analysis (Ref. (5). (5). Clearly these low levels would be unacceptable in any practical sense and indicate why it is necessary to call on the turbulent mixing in the intermediate and far wake regions tohelp increase the centerline velocity and improve the power production situation. (7b) Here. uv . For current purposes. Figure 6 provides the resulting power ratios predicted using Eqs.e. in general. Not surprisingly. For the purpose of demonstrating the current model in a generic fashion. Using this as a means of estimating Xm. However. full size wind turbines located in wind farms both on and off shore. Ri = Di/2 = DO/2. from the discussion of Section 3 above. This in turn will be used to establish X0 and ultimately couple all three regions together.92. These later results are further employed to evaluate K from Eqs. All the results are provided in Table 1 and Appendices A-D discuss each data set in detail. predicting a linear growth rate for the shear layer width. Referring to Fig. © FloDesign Inc.X ⎥ 0⎦ ⎣ D 2/3 (9a) v D = C ⎛X . the intermediate wake region begins at a point near Xi and ends at Xm. Near Xi. Figures 7 (c) & (d) provide the same results in terms of the scaled variables indicated by Eqs. 5. (7a) & (7b) as: 1/ 3 ⎛ 1/2 ⎞ ⎜ C /X ⎟ v p T ⎠ K= = ⎝ 1/ 3 1/ 2 ⎛ 2[1 . Xi is here taken to be 2 and the diameter of the near wake at Xi is given by Eq. one arrives at the relation: X m = Xi + K D 1+ u o o m D 1− u p o (10) where the value of Km has to be established from experimental data. With this in hand. 7(a) & (b) indicating a wide range of velocity defects and wake growth rates for values of X from 1 to 15. for Thrust Coefficient levels from 0. These are all provided in Figs.u ] ⎞ ⎛ ⎞ ⎜ ⎟ ⎜C X⎟ v ⎠ ⎝ ⎝ T ⎠ D D (8) The results shown in Figures 7(e) & 7(f) indicate a clear convergence of K to unity from below and above respectively as the scaled axial distances increase. the location of the virtual origin of the wind turbine’s far wake region. and Prandtl’s results apply. so that: ⎡ 1/2 ⎤ 1 ⎢ CT ⎥ uv = 1 − 2 ⎢X . and porous disc wind turbine simulators tested in a wind tunnel. 1-10 (and their reference works) covering model wind turbine rotors tested in wind tunnels. the shear layer thickness is small compared to the radial distance.In the current study. following Swain’s suggestion. 24 for example) to establish an estimate to the length of the intermediate wake region. K has been determined from a compendium of over 104 data points collected from Refs. The Intermediate Region There is no self similar solution available for this region due to its axisymmetric nature and finite diameter. (7a) & 7(b). Wilbraham MA 6 . and as indicated in Table 1.26 to 0. must be determined through coupling of the far. intermediate and near wake relations as discussed in Sections 5 & 6 below.X ⎞ ⎜ ⎟ p T⎝ 0⎠ [ ] 1/ 3 (9b) The value of X0. when the shear layer penetrates to the centerline. one can than write the more general form of the asymptotic far wake model by introducing the virtual origin. 8. one can employ Prandtl’s solution for turbulent mixing at a planar velocity discontinuity (see Ref. X0. Also. (2b). 1 at the maximum power case. the match point moves downstream as expected. CT =8/9. The Composite Wake Model Figure 9 depicts the method employed to complete the analysis. w (12a) For X>Xm. increases (decreasing mixing length). X0. Eq. Figure 10(a) shows the two components of the centerline velocity solution plus the composite solution for Km=0. (10). The most significant aspect of the velocity profiles shown in Fig. The resulting wake centerline velocity relations are: 1− u 2 o 1+ ⎢ For X<Xm. As shown in Fig. m (12c) Note the solution remains dependent on the constant. (9a) gives uw = 1− 1− u ⎡ X . (2a) gives u w = 1+ ⎡ ⎢ ⎣ ⎤ ⎥. ui and uv. 10(a) is the relatively contorted variation in the velocity as it starts with a low value at the turbine. such that the two velocities. (2b) gives D D = w p 1+ u 2u o . than decreases further quite rapidly in the near wake after where it reverses itself and begins an initially rapid but ultimately rather surprisingly slow growth toward the free stream value.X 2[1 − u ] 3 / 2 C1/2 + 1⎤ ⎥ ⎢ m m T ⎦ ⎣ ( )( ) m 2/3 . Similarly. Xm. The far wake model is coupled to the near wake by setting the virtual origin of the far wake. (10). 10(b) as the constant. (10) which is related inversely to the turbulent mixing length for the intermediate wake region. 1 + 4X 2 ⎥ ⎦ 2X (11a) For X>Xm. Wilbraham MA 7 . the resulting wake growth rate relations are: For X<Xm. As an example calculation. Eq. Eq. (11b) where 1− u 2 o ⎢1 + ⎡ ⎢ ⎢ ⎣ u m = 1+ ⎤ ⎥.X ⎞ ⎛ D D ⎞ + 1⎥ ⎜ ⎟ ⎜ ⎟ w p m p T⎝ m⎠ ⎝ m p⎠ ⎢ ⎥ ⎣ ⎦ 1/ 3 . determined from Eq. Km.D D = m p 1+ u 2u o . The case of © FloDesign Inc.6. (9b) gives where 3 ⎤ ⎡ D D =D D ⎢C ⎛ X . 2 ⎥ 1 + 4X ⎥ m⎦ 2X m (11c) and Xm is given by Eq. Km of Eq. are equal at the match point. (12b) . Eq. 11 (b).99. the variation in the intermediate wake is seen to be one of the controlling elements of the far wake growth. 13(b).26 to 0. 12 again highlight the three embedded trends: a rapidly expanding near wake. aside from the value of Km.6 to 0. they all suffer from an initial deficit and recover very slowly toward the free stream value.2 t0 0. One has to be over six diameters downstream to see reasonable velocity recovery which does not begin to attain the 90% level until nine or more diameters distance. most notable is the very large variation in the wake structure for the typical range over which one expects a HAWT to operate. the dependency of the thrust coefficient on the wind speed in Eq. In Fig. an infinitely large Km corresponds to a mixing length of zero and would thus produce an infinitely long intermediate wake length. in order to obtain accurate predictions for a particular configuration and wind environment. Fig. (11) & (12) contains only one parameter. Its verification. the data sets shown in Fig. Summary and Conclusions A first-principles based. Seen in this form. expose their embedded three trends. Thus the data sets of Table 1 and Appendices A-D. especially when one considers the wide range of measurement conditions and techniques employed. Km was than varied to achieve a visually good fit. These are given in Table 2. the power degradation ratios. 7. shown in Fig. which in this case was found to be with Km = 0. The resulting comparisons are provided in Figs. CT. i. 13 provides a summary of results of the current analysis for a range of thrust coefficient.92. Most interestingly. it is seen that the presence of the near and intermediate wake is essential to setting up the proper approach to the far wake region. as shown in Figs. established herein through comparison with a new and large compendium of measured results. Here. P2/P1. 11 consistently track the measured values and. Finally. Even though the velocities asymptotically merge rather quickly in the far wake. 13(a). (c). In particular. CT. Solutions were generated for the velocities and wake growth rates using Eqs. To obtain an estimate to the applicable value of Km for wind turbines. (e) and (f) where the low values in the near wake region near L = 2 are well represented and seen to set the basis for the relatively low values measured at L = 7 and 9 The wake growth results shown in Fig. as such. 13 (c) are also very worrisome because of the low values of 0. The flow structure and performance is clearly dominated by the sometimes large velocity trough in the near wake seen in Fig. the analytical results show consistently good qualitative agreement with the data trends and remarkable agreement quantitatively for all conditions. Overall. the wake never exceeds 2. an intermediate wake whose length shrinks as CT increases and a far wake growth rate that increases as CT increases. The model employs two “universal constants’ related to Prandtl’s turbulent mixing length. Similarly. (5) for each CT level. that characterizes the wind turbine’s influence on the flow structure. 7 were again employed. The analytical wake velocities shown in Fig. (11) and (12) and the power degradation ratios using Eq. In this regard. (5). it is first worthy of note that.e. In should be borne in mind that these predictions can only be viewed as a guide and that the model must be adapted to the particular HAWT performance characteristics. even up to L=10.5 turbine diameters. which would be difficult to decipher without the model’s guidance. the composite model of Eqs.1.7 produced for nearly all but the lowest thrust levels. These wake growth predictions do not indicate row to row interference for Turbines 2 for up to 10 diameters and well beyond for a typical HAWT lateral spacing of W = 5 or more. It is most satisfying to see that all the data sets do appear to strongly reflect elements of the three tiered structure employed in the current analysis. closed form analytical model is now available for estimating the impact of a wind turbine on its immediate down-wind neighbor. This is also especially relevant to the prediction and data comparisons of the power degradation ratio. from 0. 11 and 12 for CT ranging from 0. Wilbraham MA 8 . The model contains three critical elements that should be accounted for and accommodated in all future wind turbine wake analyses efforts: © FloDesign Inc. were first grouped into eight CT levels about which CT varied by less that 5%. the wake growth rates show a much higher sensitivity to the thrust level and are worrisome because of their implications for possible interference with the ground plane for a typical HAWT tower height of 1 to 1. these results indicate two additional trends: the wake width is very sensitive to CT and even a distance of 16 diameters down-wind. P2/P1.Km = 0 corresponds to an infinitely large mixing length and thus an intermediate wake region length of zero..5 diameters in width. provides a new direction and pathway for addressing the critical issue of wind farm design layout and productivity. the latter being very near stall. and Gyatt. Vol. pp.T. P. A. 2006.C. Hansen. and Masson.. Sheerin..W. 8. A model of the essentially inviscid near wake flow regime that is critical to setting the initial conditions for the development of the downstream flow structure. M. 2008. It is also apparent that there is a need for more and more precise measurements of wind turbine wake centerline velocities for further verification and refinement of the model. Frandsen. it cannot be controlled or overcome. R..G. and Barthelmie. A.. pp 147-167. Rethore. 1999. 27. they can suffer significant wake induced productivity losses. Journal of Wind Engineering and Industrial Aerodynamics. “Wind turbine wake aerodynamics”. M. Mechali. 96.1. Near-wake behavior of wind turbines”. Jensen. pp 466-477. 2..M.. “Energy Effectiveness of Arbitrary Arrays of Wind Turbines”. Sorensen. and Sorensen.. Barthelmie.P. M.T. 10. and Frandsen. F. K. presented at European Wind Energy Conference. The Science of Making Torque from Wind. “Wake effects at Horns Rev and their influence on energy production”. Folkerts. Citavecchia. Vermeer. Vol. J.. Nov. N. Critical Issues in the Design and Assessment of Wind Turbine Arrays”. R. 6.. P. 11... Italy.. “Modeling and measurements of wakes in large wind farms”.F.. M. and Smoroto. presented at OWEMES Conf.J. R. For longitudinal spacing of less than 10 diameters. S. With this model in hand.J. Prospathopoulos.. 3. S. L. Mechali. September 1981.. J. The model elements for the near and intermediate wakes of Turbine 2 would be essentially the same as that of Turbine 1 adjusted accordingly. Phillips. Wilbraham MA 9 . Barthelmie. Politis.E. Schepers.. The fundamental root cause of this is the deep velocity trough encountered aft of the lead turbine. © FloDesign Inc. Lissman. and Rethore. 467-510. E. P. “Calculating the Flowfield in the wake of wind turbines”.. “Modeling and measurements of offshore wakes”. “Three-Dimensional Wakes of Simulated Wind Turbines”. Vol. “Wake Modeling for intermediate and large wind farms”. R. S. A methodology for estimating the length of the intermediate wake and thus the effective starting point of the far wake. paper BL199 presented at EWEC Wind Energy Conference Conf. Neubert. A. Eechen. Vol. Journal of Physics Conference Series. References 1. and van der Piji. Rathmann.. As such. S.J. Schlez.. Jensen. attention can now be turned to its generalization for third and beyond rows in a wind farm. Vol. J. Vol. J.. .M. Stockholm. The model verifies and reinforces what has begun to become apparent in recent studies of HAWT wind farms. Vol. “Offshore Wind Turbine Wakes Measured by Sodar”.. 3. 1988. C. 12. 6. 2.. December 21-24. Frandsen. pp 103-102. El Kasmi. S. Barthelmie.. R.. Frandsen. AIAA Journal. No. “.T. “An extended k-ε model for turbulent flow through horizontal-axis wind turbines” Journal of Wind Engineering and Industrial Aerodynamics. 4.. P. P. K. P.. Athens Greece. Journal of Wind Engineering and Industrial Aerodynamics.. R..N. Sforza. 19.. October 2006 Barthelmie. Frandsen. L.. 2006 Rathmann. P. pp 213-224. Vol.B.. and Nielsen. A far wake model based on the Prandtl/Swain analysis and scaling. No. Zalay. 7..-Dec. 2006 Rathmann. 9. P.E. L. 20. J. presented at EWEC Wind Energy Conference Conf. 2007 Barthelmie. 2003. pp 323-327. One approach that should be considered would be to combine the effects of the Rows’ 1 & 2 thrust deficits into a single element influencing the Prandtl/Swain far wake development approaching the third row and so on. O. Ormel. Presented at the International Symposium on Wind Energy Systems. April 2223.. S.J. Sanderhoff. O. S... published in Progress in Aerospace Sciences. Rados. A. 3. L. Sweden.. O.. 1979 Lissman. 5..S. 39.S.J. it should be accounted for in all future cost analyses of wind farm output potential using the specific performance characteristics of the HAWT employed to tune the model accordingly. W..B.. “Turbine wake Model for Wind Resource Software”. 2007. D. Magnusson. pp 1101-1107. Ainslie. Because this is a result of the basic aerodynamics of HAWTs.J. Stobbe. G.. the impact of which persists into the far wake. 80.. 13.. Journal of Atmospheric and Oceanic Technology. O.. P. and Crespo. Cabezon. 1982. 75.. G. Journal of Energy. Pryor... 9. 2005... M. LTD. N. 25. 1926 and in Aerodynamic Theory p 166 by W.J. March 1-2. Theory. H. H. J. L.G... Seifert. H. loback-Travemunde. Vol. June 2007 © FloDesign Inc. and Lumley. McCormick. 2004. Barthelmie. and Twele. 27. Larsen. Gothenberg. 20. “The mechanics of viscous flows”. Katic. “A Multi-turbine Power Curve Approach”. Lange. MIT Press. L. Wind Turbines. “Ducted Wind/Water Turbines and Propellers Revisited”. Germany. „Results from a joint wake interference research program“. Schomburg. B. pp 647-659. Durand. Swain. Larsen. Dover Publications.. J. Rome.. M. F. G. Technical Note accepted for publication in the AIAA Journal of Propulsion and Power. Rados. “Comparisons of Wake Models with data for Offshore Windfarms”.. 2003. AERODYNAMICS OF V/STOL FLIGHT. 1960..J. Zurich.. Werle. S. Boundary Layer Theory. A First Course in Turbulence. S. Barnes W.. and Jensen.. Wind Engineering. 2001. 2002 23. Sweden. Chamlers University of Technology. G. Barthelmie.. June 17.M. November 1.. 24. O. 9 pp 39-53.. Schlichting. Hegberg. Eric. Madrid. Waldl.. M.. presented at the European Community Wind Energy Conference.. M. Hojstrup.. Presented at the Nordic Wind Power Conference. West Sussex England..14. No. K. Series A. 1929. 2006. Minneola NY. Manwell. Pryor. Wind Energy.. Presented at the European Wind Energy Conference. A. R. Jr. and Magnisson..O... Hau. Kramkowski.. Albers. Schepers. T. 18. I. and Holttinen. Presented at European Wind Energy Conference. Beyer. J. and Presz. U. and Kronig. “Analytical Modeling of Wind Speed Deficit in Large Offshore Wind Farms”.. Prandtl. 19. Ratmann. . Schlez. Frandsen. 1999 28. Vol. And de Witt. “On the Turbulent Wake Behind a Body of Revolution” Proceedings of the Royal Society of London. J.F. S. 17... 22... W. March 8-12. 1935..L. R. Design and Application. Second Edition. 125. 2002 21. Tennekes.. Solarpraxis. Norgaard. No. and Thorgersen. A. W. Spain. “A Simple Model for Cluster Efficiency”. 25. H. Jr. Wilbraham MA 10 . 1972 26. October 7-9. Schlez.. Springer Publishing Company. appearing in the Proceedings of the 2nd International Conference of Applied Mechanics.G. 15. John Wiley & Sons. R. 1993 16. J. J. 1986. P. McGowan.. T. Switzerland.L.799. H. Fourth Edition. Gashe.. “Recommendations for Spacing in Wind Farms“. and Rogers. H. Vol. WIND ENERGY EXPLAINED.. Italy. J.. Hojstrup. p 62. W. 9 pp 271280. Schield.: WIND POWER PLANTS.M.. A. McGraw Hill Book Company. 340 0.590 1.540 0.820 0.5 11.634 0.7 0.4 8.379 0.691 1.890 0.6004 1.560 9.105 1.65 8.927 0.840 0.0 8.265 1.926 1.840 0.817 1.603 1.714 0.265 1.333 1.5 6 7.768 3.714 0.5 6 6 6 6 8.615 0.290 1.430 0.167 1.424 0.539 0.4 9.780 0.427 1.451 0.0-9.0 8.5 10 2 5 7.0-8.799 10.163 1.5 3.903 0.809 0.789 0.8 3.0 8.4 3.5 7 7 7 7 9.701 0.744 0.519 1.5 11.0 8.575 0.430 0.301 1.707 1.5 12 8.912 0.4 9.94 5.600 0.249 1.26 0.662 0.118 0.939 0.500 8. Wilbraham MA 11 .94 5.295 1.430 0.0-9.65 7.643 0.0-9.973 Wind Tunnel Rotor Test Wind Tunnel Rotor Test Wind Tunnel Rotor Test OnShore Wind Farm OnShore Wind Farm OnShore Wind Farm OnShore Wind Farm Wind Tuneel Porous Disc © FloDesign Inc.498 0.688 1.570 0.432 0.840 0.4127 0.5-12.132 1.700 0.3014 0.775 1.284 1.641 0.930 0.26 0.820 0.913 0.5848 0.4 9.130 7.667 0.667 0.670 0.011 2 5 7.427 0.816 0.430 0.180 1.855 0.001 1.080 7.5 7.33-9.5 6 9 12 10.737 0.670 0.451 0.5802 0.4 x/Dp 8.760 1.5848 0.127 1.716 0.860 0.5-8.728 P 2/P 1 0.4 2.5 12 5.598 8.196 1.698 0.0-9.5 9.472 1.840 0.817 1.550 0.900 7.926 7.820 0.894 0.618 0.730 0.435 0.923 0.640 0.789 1.708 0.424 0.451 0.709 0.717 0.845 1.248 5.181 0.430 0.714 0.979 1.098 8.242 1.093 1.248 1.847 0.780 0.510 7.4817 0.495 1.516 0.882 0.855 0.291 1.645 0.422 0.437 0.1602 0.923 0.294 7.560 0.71 0.883 1.695 0.760 0.5 8.370 6.870 0.876 V2 8.903 0.784 63.500 9.341 1.853 0.893 0.190 7.4 7 7 7 7 7 7 9.663 0.722 0.657 0.983 4.5 8 8.840 0.843 0.5054 1.520 4.540 2.4 9.000 11.700 0.289 1.820 0.695 0.326 1.5 9.120 8.232 1.540 5.057 2.421 0.422 0.946 0.141 6.8 6.657 0.714 0.820 0.5 9.5-12.19 8.945 2.84 10.5 6 7.6572 1.0 8.423 1.5 7.876 0.250 1.7414 0.9 7.420 0.0 7.820 0.643 0.625 0.775 0.4 8.551 1.140 7.320 1.479 0.855 0.0-9.99 7.770 0.4 10.3785 0.640 1.340 0. data) Off Shore Horn Rev (Vestas 80) WS Avg Wind Range Speed (m/s) 9 8.437 0.354 1.190 8.650 0.695 0.4 6 6 6 1.206 6.31-7.5148 0.809 0.391 1.667 0.439 0.520 7.0419 0.166 1.804 0.894 0.5-8.5-12.162 0.436 6.5 8 9.809 0.735 0.94 5.902 0.010 10.5 2.121 2.000 8.265 1.292 1.643 0.451 0.909 0.352 1.92 0.780 0.560 11.760 5.000 10.876 0.4 1.54-5.4 3.707 0.817 1.835 0.851 1.667 0.340 0.202 0.415 0.267 1.708 7.4 6.54-5.0 8.670 0.688 0.421 0.002 4.603 0.700 0.548 0.653 1.758 0.437 0.971 4.429 1.24-10.5-6.520 11.427 0.970 1.63-8.792 0.432 0.744 0.277 0.686 2.92 0.5-12.193 Ref 2 Rathmann et al 2 Rathmann et al 2 Rathmann et al 3 Rathmann et al 3 Rathmann et al 3 Rathmann et al 3 Rathmann et al 3 Rathmann et al 3 Rathmann et al 3 Rathmann et al 3 Rathmann et al 4 Barthelmie et al 4 Barthelmie et al 4 Barthelmie et al 4 Barthelmie et al 4 Barthelmie et al 5 Mechali et al 5 Mechali et al 5 Mechali et al 5 Mechali et al 5 Mechali et al 5 Mechali et al 5 Mechali et al 5 Mechali et al 5 Mechali et al 5 Mechali et al 2 Rathmann et al 2 Rathmann et al 2 Rathmann et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 6 Barthelmie et al 7 Kasmi et al 7 Kasmi et al 7 Kasmi et al 7 Kasmi et al 7 Kasmi et al 7 Kasmi et al 7 Kasmi et al 7 Kasmi et al 7 Kasmi et al 8 Magnusson 7 Kasmi et al 8 Magnusson 7 Kasmi et al 7 Kasmi et al 7 Kasmi et al 9 Ainslie 9 Ainslie 9 Ainslie 9 Ainslie 9 Ainslie's Ref 11 9 Ainslie's Ref 11 9 Ainslie's Ref 11 9 Ainslie's Ref 10 9 Ainslie's Ref 10 9 Ainslie's Ref 10 9 Ainslie's Ref 10 9 Ainslie's Ref 13 9 Ainslie's Ref 13 9 Ainslie's Ref 13 9 Ainslie's Ref 13 9 Ainslie's Ref 13 9 Ainslie's Ref 13 9 Ainslie's Ref 12 9 Ainslie's Ref 3 9 Ainslie's Ref 3 9 Ainslie's Ref 3 9 Ainslie's Ref 3 9 Ainslie's Ref 3 9 Ainslie's Ref 3 10 Sforza 10 Sforza 10 Sforza 10 Sforza 10 Sforza 10 Sforza 10 Sforza 10 Sforza 10 Sforza 10 Sforza Off Shore MiddleGrunden (Bonus 76) Off Shore Vindeby Off Shore (Nibe B) Off Shore (Danwin 180) 38.3964 0.540 0.118 0.324 1.817 1.901 0.800 0.800 0.843 0.784 137.5 2.438 0.432 0.688 0.6268 0.614 7.5 1 1 1 4.340 0.189 1.987 6.946 1.800 0.424 0.098 7.748 1.505 0.695 0.946 0.730 0.672 0.0 7.760 0.5 6 7.4 2.5 8.5 11.520 11.5905 0.33-9.428 1.523 0.526 0.4 1.784 0.63-8.740 6.695 0.5936 V2/Va 0.093 1.280 0.5 12 8.316 1.451 1.699 1.439 CT 0.990 5.107 2.796 0.432 0.392 0.560 7.670 0.780 0.5183 1.660 10.901 0.427 0.983 3.7 2.835 0.5-9.0 8.61 0.222 6.700 0.880 11.516 2.0-10 9.340 0.4 7 7 7 2.876 0.714 0.000 1.073 1.243 63.760 8.670 0.760 0.670 0.38-6.712 1.559 0.0-9.798 1.86 6.4 5 2.463 0.670 0.770 0.5 12 8.54-5.919 0.700 0.4 6 6 6 6 6 6 8.26 0.6 4.0 7.784 111.843 0.070 7.324 1.714 0.909 0.19 8.575 1.741 8.094 1.230 1.532 0.971 4.770 0.695 0.161 1.5 8.92 L 9.540 0.4 9.944 2.5-8.09-7.5-10.630 0.12-6.2 2.624 0.500 4.62 6.640 0.987 6.817 1.700 0.914 1.835 0.340 0.882 0.Table 1: Wake Database (yellow highlight implies calculated from Ref’d.767 0.000 8.928 2.5 10 12.422 0.297 0.000 P1 (MW) 0.92 0.7792 0.714 0.5 8 7.070 7.4026 0.560 9.5 4.975 1.721 1.5 0.540 8.796 0.559 0.640 0.780 0.5 10 7.820 0.958 9.15 9.607 0.662 0.432 0.118 0.614 1.817 1.49 6.344 P2 0.5105 0.8007 0.4 7 7 7 9.585 0.664 1.819 0.825 0.470 0.888 0.809 0.1803 0.995 3.5 9 8.0-8.5 11.405 2.0-10 9.0-9.606 1.870 0.840 5.92 0.834 0.506 0.640 0.776 0.540 0.659 0.987 2.800 0.559 0.5 15 Dw/Dp 1.160 1.4012 0.748 0.695 0.1 2.0-10 9 7.5 12 8.740 5.621 6.0-10 9.0-8.5-9.216 63.043 6.250 10.605 1.817 1.26 0.400 0.4 8.5-12.740 5.799 0.677 0.5 6 7.500 8.762 8.775 2.650 0.5591 0.499 1.890 0.435 0.432 0.4 8.204 1.4 6 6 6 8.794 7.884 CP 0.901 0.92 0.451 0.432 0.259 0.504 0.864 0.770 0.5 11.903 0.4 8.790 0.310 11.540 0.809 0.969 1.353 7.4 1.4 9.487 1.021 2.820 0.000 8. 703 0.634 0.971 3.4 1.650 0.840 0.780 0.643 0.670 0.643 0.4 10.5 © FloDesign Inc.714 0.540 0.8 2.939 CT Avg= 0.4 2.987 6 6 6 6 8.834 0.919 0.260 0.744 0.758 0.903 0.835 0.344 0.775 0.5 2.504 0.098 7.4 9.695 0.654 0.701 0.4 8.894 0. data)) P 2/P 1 uw 0.843 0.470 0.670 0.1 6 6.835 0.5 6 6 6 7.4 1.667 0.167 1 0.707 0.792 0.451 0.451 0.667 0.971 3.5 10.451 0.819 0.659 0.714 0.4 8.618 0. Wilbraham MA 12 .645 0.700 0.505 0.002 4.909 0.876 0.451 0.4 9.379 CT 0.767 0.663 0.987 5 6.983 3.987 6.670 0.799 0.4 2.700 0.Table 2: CT Correlated Database (yellow highlight implies calculated from Ref’d.540 0.4 8.760 0.709 0.4 2.744 0.799 7.662 0.641 0.990 5.853 0.4 9.923 0.695 0.893 0.888 0.615 0.770 0.677 0.451 0.979 4.260 0.923 0.260 0.876 0.695 0.695 0.902 0.643 0.695 0.876 0.340 0.804 0.4 0.714 0.540 0.640 CT Avg= 0.662 0.698 0.260 0.5 2.770 0.695 0.640 0.4 0.610 0.451 0.540 0.667 0.710 0.770 0.716 0.894 0.912 CT Avg= 0.4 4.670 0.882 0.840 0.714 0.901 0.625 0.011 4.780 0.667 0.4 1 6 6 6 8.882 0.4 8.260 0.714 0.162 0.4 6 1.708 0.695 0.748 7 7 7 7 9.043 6 6.927 0.670 0.4 2.820 L x/Dp 2 5 7.735 0.714 0.913 0.780 0.540 0.670 0.4 1.4 7.780 0.4 7 7 7 9.600 0.700 0.657 0.890 CT Avg= 0.5 10 7 2.717 0.903 CT Avg= 0.864 0.700 0.670 0.700 0.548 0.500 0.5 2.550 0.901 0.657 0.585 0.640 0.4 7 7 9.539 0.768 6 6 6 8.796 0.890 0.770 0.730 7 7 7 9.607 0.737 0.780 0.870 0.909 0.650 0.542 0.700 0.9 2.181 7 7 7 9.789 0.516 0.520 4.4 9.903 0.570 0.860 0.5 8.2 6.714 0.847 0.057 3.479 0.640 0.4 9.870 0.624 0.760 0.730 0.4 6 6 8. For the Vendeby site. in the current analysis. the velocities in the wake aft of the first wind turbine were inferred from the second wind turbine’s power output and its related power curve.5 Power (MW) 1. This limits the data to a few values of the second turbine’s power at a single distance L. Figure A-1 provides the measured power curves for both the Vestas 80 and Bonus 76 wind turbines. the velocities in the wakes were measured by a Sodar system and adjustments to the wake locations were not necessary.8 Bonus 76.Appendix A: Horn Rev.0 1. For the Horn Rev and Middlegrunden sites. CT(P) thrust measured Thrust Bonus 76.e. only the data downstream of the first turbine was employed. 1-6.2 0. Some of Refs. only those where the two turbines were aligned to within 1% to 2% of the wind axis were used. For current purposes.5 Vastas 80 Vestas 80 Bonus 76 Bonus 76 Bous 36 Bonus 36 2. Coeff. 1-6 provide the measured power output of the second turbine (and others) while some provided the inferred effective wind speed for that power output. MiddleGruden & Vindeby The data of Table 1 for the three off-shore sites at Horn Rev (80 2MW Vestas 80 wind turbines).4 0.0 0. Coeff. Middlegruden (20 2MW Bonus 76 wind turbines) and Vindeby (11 450kW Bonus 36 wind turbines ) employed herein are provided in various forms in Refs. i. 80.5 0. In an attempt to account for the upstream influence/disturbance induced by the second wind turbine on its approaching velocity field. The power curve for the Bonus 36 was taken here as an area adjusted version of the Bonus 76.6 CT 0. these inferred effective wind speeds were assumed to be representative of the velocity one diameter ahead of the second turbine. Pmeasured power 0. 2.0 0 5 10 15 20 25 Wind Speed (m/s) 0. For the current study. Wilbraham MA 13 . CT Measured Thrust measured power Vastas measured Vestas 80. its distance downstream from the first turbine was reduced by one wake diameter.0 0 5 10 15 20 25 Wind Speed (m/s) Fig A-1: Power Curves and Thrust Coefficients © FloDesign Inc. as provided in Ref. This is the source of the significant difference in the inferred levels of outlet velocity shown in Fig.8 uo.0 0. for all three turbines based on the measured data power curves of Fig A-1. A-2. ⎜ P ⎟ ⎥ ⎢ ⎜ ⎟ 3 32 ⎠ ⎝ ⎦ ⎣3 (A-1) where: CP ≡ P 1 ρA V3 2 p a .6 C P .0 Bonus 76 & 36 uo. (4b) to write that: 8C CT = ⎤ ⎡ P cos 1 A cos⎛ . Wilbraham MA 14 .6 0. which were found to be operating at approximately an 80% level.2 0. thrust measured 0.0 0 5 10 15 20 25 Wind Speed (m/s) 1.4 0. (A-2) CT ≡ T 1 ρA V 2 2 p a .2 CP . 1. Also shown are the calculated thrust coefficient values obtained from the measured power levels and related power coefficient using the inverse of Eq. is reproduced in Fig A-1. thrust measured 0.0 0 5 10 15 20 25 Wind Speed (m/s) Fig A-2 Vestas and Bonus Performance Parameters © FloDesign Inc.The Bonus 76 measured thrust coefficient.C1/2 27 ⎞ + 240o . 1.8 Vestas 80 Outlet Vel uo 0. Figure A-2 provides the power coefficients and outlet velocity. The differences in the measured and calculated power coefficients and thrust coefficients occur because of inherent aerodynamic and mechanical inefficiencies in the respective wind turbine systems.4 Vestas 80 Power Coeff CP 0. (A-3) and Va is the effective undisturbed wind speed approaching the turbine. uo. power measured 0. power measured 0. Following Refs. 2 u (1 − u ) w w C (A-5) © FloDesign Inc. ⎜ w p⎟ ⎝ ⎠ 0 (A-4) Which can be integrated to determine an effective wake width by assuming the wake velocity is constant at its centerline value. leading to the relation: D w D = p T .u)d⎛ D /D ⎞ . Wilbraham MA 15 . 1-6. the values of the wake widths shown in Table 1 were calculated from momentum and mass conservation along the axial direction assuming a constant pressure state to write that: ∞ 2 CT = ∫ u(1 . 0 uo.2 CP . Also.Appendix B: Nibe B and Danwin 180 The data presented in Table 1 for these two wind turbines was reproduced from Refs. Additionally. the differences between the measured and calculated thrust coefficients is an indication of the system operating at approximately 80% efficiency. thrust measured 0.0 6 8 10 12 14 16 Wind Speed. power measured 0. For current purposes.4 0. Wilbraham MA 16 . Eq. which themselves took the data from reports presented in 1985 and 1996. 8 and is reproduced below with all the attendant performance parameters. the power curve for the Danwin 180 was provided in Ref. Danwin 180 200 150 Power (kW) 100 50 0 6 8 10 12 14 16 Wind Speed (m/s) 1. power measured 0.6 CT. A-5 was used to calculate the effective wake diameter. © FloDesign Inc. The results were presented in terms of the velocity profile across the entire wake over a range of measured wind velocities. 7 & 8. power measured 0. (m/s) Fig B-1: Danwin 180 Parameters As in Appendix A.8 CT. measured thrust levels and wake distances. only the centerline or nearest-to-centerline velocities were employed for consistency with the other data sets. 9) CT=0.80 Field Rotor (Ref. 9) CT=0. 9) CT=0.13 of Ref. the values of the effective wake diameters were calculated using Eq. 10 were averaged and recorded in Table 1.12 of Ref.14 of Ref. 9. Ainslie presents measured wake centerline velocities for a range of configurations from his own organization plus from other organizations and authors. A-5 1.74.10 of Ref. 10 presented in Table 1 was acquired in a wind tunnel using a porous disc model to simulate the turbine effect. wind turbine field tests and wind tunnel actuator disc tests.31. calculated from the stated thrust coefficients.70 Field Rotor (Ref. however have been excluded from the current set given in Table 1 due to anomalous patterns observed in the data.10 of Ref. 9) CT=0.10 of Ref. Ref.54 Field Rotor (Ref.1 1 10 x/Dp Figure C-1 Data From Ref. 9) CT=0. Turb=0. Ansilie also noted the different behavior for these two cases but indicated that it was possibly due to the low level of turbulence. 9) CT=0. Two case.13 of Ref. © FloDesign Inc. Wilbraham MA 17 . 9) CT=0.67 0. Turb=2% Tunnel Rotor (Ref. Because the data was taken in the near the wind tunnel floor to simulate surface boundary layer effects.0 uo uo uw Tunnel Disk (Ref.64 Field Rotor (Ref. the measured lateral and vertical wake diameter values provided in Ref. These are shown in Fig C-1 for the top two cases of the legend on the chart (the very low turbulence level cases) where it is observed in the figure that the measured wake velocities are below the outlet velocities.60 Field Rotor (Ref. C-1. 9 Appendix D: Sforza Porous Disc Wind Tunnel Tests The data of Ref.84 Tunnel Rotor (Ref. 9) CT=0. 9 presented measured results for rotors in wind tunnels. the wake was found to grow asymmetrically in the vertical and lateral directions. uo. 9) CT=0. This result is inconsistent with momentum deficit considerations and implies either the reported thrust levels are too low or the velocities reported are in error for these two cases. Again.3 of Ref. 9) CT=0.5% Tunnel Rotor (Ref. all of which are shown below in Fig.78 Tunnel Rotor (Ref.Appendix C: Ainslie Results In Ref. For current purposes. Wind Farm Array V = Va p = pa Capture Streamtube Mixing Layer p increasing Near Wake Intermediate Wake p > pa p = pa Far Wake Fig 2. Wilbraham MA 18 . HAWT Wake Flow Structure © FloDesign Inc.Row 2 L=l/Dp W=w/Dp Va Row 1 1 2 3 Row 3 Fig 1. 8 CT=0.6 CT=0.6 -5 -4 -3 -2 -1 0 1 2 3 4 5 X 4. Max Power CT=0. Max Power 0. Stall 1.6 ui 0. Stall 3 4 5 X 1.6 CT=0.0 CT=0.2 0. Inviscid Model © FloDesign Inc.8 (b) Capture Streamtubes 0.2 0.8 1.2 Di/Dp 1.0 CT=0.2 1.0 -5 -4 -3 -2 -1 0 1 2 CT=1.0. Wilbraham MA 19 .8 0.Capture Streamtube ui=1 ui=1 up pu pd Dp Di uO x Fig 3. Inviscid Flow Model Nomenclature 1.0.4 CT=8/9.4 CT=8/9.6 CT=1.2 (a) Velocities 0.4 CT=0.4 CT=0. Wilbraham MA 20 . Stall CT=0.6 0. Max Power CT=0.4 CT=0.0 0.2 CT=8/9.0. Typical HAWT Power and Thrust Curves 1.4 CT=1.0 1 2 3 4 5 L Fig 6.P delivered to generator P1 CT Measured CT P2 V2 Va Relative Wind Speed Fig 5.8 CT=0. Inviscid Power Ratios © FloDesign Inc.6 P 2/P 1 0.2 0.8 0. 6 Dv/Dp 2.0 (e) Velocity Deficit Constant 0.10 Dv/Dp 2. 10 3.1 0 4 8 12 16 1/2 X/CT CTX Fig 7. 9 Field Data. Ref.1. Refs.0 (b) Wake Growth 2.2 0. Refs. 12-14 of Ref.0 1-uv 0. Ref. 8.0 0 4 8 12 16 20 1. Refs.5 0.0 12 16 20 0 4 8 12 16 CTX 10.5 (d) Wake Growth 1-uv 0. 2-5 Off Shore. 10-11 of Ref.4 (a) Velocity Deficit 1.00 3. 9 Wind Tunnel. Intermediate Wake Structure © FloDesign Inc. Porous Disc. 2 Off Shore. 6 Off Shore.5 0.01 0 4 8 1/2 X/CT 1.0 1.8 Off Shore.0 10.1 0 4 8 12 16 20 (f) Wake Growth Cnstant 0.0 K 1.0 0 4 8 12 16 X 1. 3.5 0. 7-8 Wind Tunnel. Wilbraham MA 21 . Ref. Wake Data Ro=Do/2 Rp=Dp/2 Xi Xm L Fig.0 0. Refs.0 K 1.0 X (c) Velocity Deficit 2. 0 Km =0.0 0. Model Components and Mixing Length Influence at Max Power © FloDesign Inc.4 Km =0. 10.0 0. Max Power 0.1 0. ui=uv uv ui uO X Xm Fig.2 Viscous Soln (a) Km=0. Max Power 0. 9.1.0 0 2 4 6 8 10 X Fig. Composite Wake Solution Structure 1.8 Km =0.6 uw 0. CT= 8/9.uw up um X0 Match Point. CT= 8/9.4 0. Wilbraham MA 22 .8 Composite Soln 0.2 0.2 (b) Km Influence.6 uw Inviscid Soln 0.0 0 2 4 6 8 10 X 1. Data uw .8 0. L0=1 © FloDesign Inc.8 uw & P 2/P 1 0.2 uw . Km=0. Model P 2/P 1.2 (e) CT=0. Model P 2/P 1.8 uw & P 2/P 1 0. Model P 2/P 1.1. Data uw .260 0.0 1.0 X&L 1. Data P 2/P 1. Model 0.0 0.0 1.0 0 4 8 12 16 0.0 0 4 8 12 16 0. 11. Model P 2/P 1.2 0. Data 0. Model 0. uw & P 2/P 1 0.6 uw .451 0.8 uw & P 2/P 1 0. 0. Data P 2/P 1.6 uw .0 0 (f) CT=0. Velocity & Power Data and Model Comparisons.4 0.2 (c) CT=0.838 0.4 0.340 4 8 12 16 0. Wilbraham MA 23 . Model 0. Model P 2/P 1.4 uw & P 2/P 1 0.920 8 12 16 X&L X&L Fig.4 0.8 0.6 uw .703 4 8 12 16 X&L 1.4 uw .4 uw & P 2/P 1 0. Model 0.2 0. Data uw .654 0.0 1.2 0.6 uw .6 uw .0 0. Data P 2/P 1.0 0 (d) CT=0. Model P 2/P 1.6 & P 2/P 1 0.1. Model P 2/P 1. Data uw . Data uw .8 0.4 uw .2 (g) CT=0.0 X&L 1.6 uw .0 0 4 (h) CT=0. Data uw .0 X&L 0. Data 0.2 (a) CT=0.0 0 4 8 12 16 0.0 0. Model 0.8 uw & P 2/P 1 0.6 uw . 0. Data P 2/P 1.8 0. Model P 2/P 1.542 4 8 12 16 X&L X&L 1.0 0 4 8 12 16 0 (b) CT=0.4 uw 0. 5 X 2.0 1. 12.5 (e) CT=0.0 0 0 4 8 12 16 (b) CT=0.5 2.0 0.703 0.0 Dw /Dp 1.5 0.0 0.5 Dw /Dp 1.0 8 12 16 0 4 (h) CT=0. Wake Growth Data and Model Comparison.5 1.1 © FloDesign Inc.5 Dw /Dp 1.0 X X 2.5 Dw /Dp 1. Wilbraham MA 24 .0 Dw /Dp 1.260 0.5 2.5 2.5 2.451 0.0 Data Model 1.0 Data Model 2.0 0.838 0. Km=0.920 8 12 16 X X Fig.2.0 Data Model 1.0 0.0 0.654 0.340 4 8 12 16 0.5 2.5 Data Model 1.5 0.0 1.5 Data Mode 2.0 0 4 (g) CT=0.5 2.0 0 4 8 12 16 (f) CT=0.0 0 4 8 12 16 X X 2.0 Data Mode 1.5 (c) CT=0.0 0 4 8 12 16 0.5 Dw /Dp 1.542 8 12 16 X 2.0 Dw/Dp 1.5 Dw /Dp 1.0 0 4 (d) CT=0.5 (a) CT=0.5 1.5 0.0 Data Model 2.5 2.5 0.0 Data Model 2. 8 C T=0.2 0.2 C T=8/9 CT=0.6 T C T=0. 13.2 CT=0.1.4 CT=0.6 0.4 C T=0.0 CT=0.0 2.2 1.0 0 2 4 6 8 10 X 1.0 0.8 0.2 0.4 C T=0. L0=1 0.8 CT=0.6 0.5 C T=0.4 C T=0.5 CT=8/9 CT=0.8 0.6 uw C T=0.99 0.6 P 2/P 1 0.0 (c) Power Ratios. Model Predictions for Km=0.99 6 8 10 L Fig.99 2.8 C =0.0 0 2 4 6 8 10 X 3. Wilbraham MA 25 .5 (b) Wake Growth 0.4 CT=8/9 (a) Wake Velocities 0.0 0 2 4 CT=0.0 Dw /Dp 1.1 © FloDesign Inc.