Performance of an H-Darrieus in the Skewed Flow on a Roof 1

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1 Sander Mertens Gijs van Kuik Gerard van Bussel Delft University Wind Energy Research Institute, Faculty of Civil Engineering and Geosciences, Stevinweg 1, 2628 CN Delft Performance of an H-Darrieus in the Skewed Flow on a Roof 1 Application of wind turbines on roofs of higher buildings is a subject of increasing interest. However, the wind conditions at the roof are complex and suitable wind turbines for this application are not yet developed. This paper addresses both issues: the wind conditions on the roof and the behavior of a roof-located wind turbine with respect to optimized energy yield. Vertical Axis Wind Turbines (VAWTs) are to be preferred for operation in a complex wind environment as is found on top of a roof. Since the wind vector at a roof is not horizontal, wind turbines on a roof operate in skewed flow. Thus the behavior of an H-Darrieus (VAWT) is studied in skewed flow condition. Measurements showed that the H-Darrieus produces an increased power output in skewed flow. The measurements are compared with a model based on Blade Element Momentum theory that also shows this increased power output. This in contradiction to a HAWT in skewed flow which suffers from a power decrease. The paper thus concludes that due to this property an H-Darrieus is preferred above the HAWT for operation on a flat roof of higher buildings. DOI: / Introduction Apart from the use in remote areas, the use of small wind turbines for the built environment has gained in interest. This originates from two sides: the growing demand for renewable energy and the increased interest among architects, project developers and local governments in sustainable and zero- or low-energy design buildings. The interest in sustainable, zero- or low-energy design buildings is caused by, among other reasons, more stringent rules on and the awareness of the high-energy consumption of buildings. In The Netherlands, the demand for renewable energy has been raised because of two main reasons see Scheepers et al. 1 and Greenprices 2: Europe signed the Kyoto protocol that demands the reduction of the carbon dioxide emission. This resulted in national renewable electricity production goals On the 1st of July 2001 a liberalized renewable electricity market was introduced. This resulted in a large demand for renewable energy. According to Panofsky and Dutton 3, the roughness of the built environment reduces the average wind speed at about 75% of the average building height to zero. As a result, wind turbines in the built environment require buildings that are reasonably higher than the average building height. Wind turbines close to such buildings can profit from the local flow deviation since they operate in wind accelerated by the building. Buildings may play an active role in increasing the energy output and are not just used as a foundation of the wind turbine. Such integration of building and wind turbine will therefore be referred to as Building Augmented Wind Turbines BAWTs. There are three different categories of BAWTs: those situated on a building; those situated between diffuser shaped buildings; and those situated in a duct through a building see Mertens 4. The first category can be explored in the near future; suitable 1 Copyright 2003 by Delft University of Technology. Published by the American Institute of Aeronautics and Astronautics, Inc. and the American Society of Mechanical Engineers with permission. Contributed by the Solar Energy Division of THE AMERICAN SOCIETY OF ME- CHANICAL ENGINEERS for publication in the ASME JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received by the ASME Solar Energy Division, December 19, 2002; final revision, July 17, Associate Editor: D. Berg. wind turbines are being designed but the buildings exist. The last two categories are interesting for the far future because both building and wind turbine should be designed almost from scratch. The use of an H-Darrieus wind turbine on buildings seems a favorable proposition. The H-Darrieus does not suffer from frequent wind direction changes, architects like the design and the integration of it in the building, and furthermore, as this paper will show; the performance coefficient of an H-Darrieus can be higher than the performance coefficient of a HAWT in similar conditions. More specifically, this paper concerns an H-Darrieus in the skewed flow on a flat roof of a mid- to high-rise building. Of course the wind turbine could be tilted towards the average flow, making normal operation possible, but this requires an expensive tilt mechanism and knowledge of the expected skew angle. The influence of the direction of the flow on the performance of an H-Darrieus is described using a model based upon multiple stream tube theory, a combination of momentum and blade element theories. The performance of an H-Darrieus in skewed flow is then calculated with this model and compared with measurements of the performance of the H-Darrieus in skewed flow and calculations of the performance of a HAWT in skewed flow. When operating an H-Darrieus in rural areas, the skewness of the flow is small compared to the skewness of the flow on roofs and therefore the effect of skewness on the turbine performance can be ignored. The effect of the skewed flow on the performance of an H-Darrieus is thus strongly coupled with the skewness of the flow on the roof, and the two topics should be discussed together. We were interested in the effect of the skewed flow on the performance of an H-Darrieus designed for siting on roofs of higher buildings, a turbine for which we performed the aerodynamic design for Core International. We carried out some tests in our open-jet wind tunnel and, to our surprise, the performance coefficient increased in skewed flow. This raised our attention and we decided to develop an analytical model in order to understand the flow phenomena behind this. The size of the skew angle is determined from a Computational Fluid Dynamics CFD calculation, and the operation of the H-Darrieus in skewed flow is discussed with help of an analytical model based on the Blade Element Method BEM and measurements in the open-jet wind tunnel of the University of Technology Delft. Journal of Solar Energy Engineering Copyright 2003 by ASME NOVEMBER 2003, Vol. 125 Õ 433

2 Skewed Flow Over Buildings The boundary layer separates at the windward roof edge of the building and the flow forms a separation bubble on the roof below the streamlines above the roof. As a consequence, the velocity vector of the flow above the separation bubble makes an angle with the horizontal roof Fig. 1. This angle is subsequently referred to as the skew angle. The skew angle is largest at the windward roof edge and decreases downwind. The local skew angle also depends on the roughness of the upwind area. For high upwind roughness, say upwind obstacles of the order of the building height, the building is barely noticeable as a separate roughness obstacle and not much mass flow is moved upwards. As a consequence, the skew angle will be small. It can be concluded that buildings relatively low with respect to the surrounding roughness objects can be associated with small skew angles and vice versa see the following section on CFD calculations. Computational Fluid Dynamics CFD Calculations This section describes 3D Computational Fluid Dynamics CFD calculations on the turbulent separation over a rectangular model building, carried out with the commercial code Fluent 6.1. The calculated streamlines are depicted in Fig. 1. The reference building for the CFD calculation has a height of 20 m, a width of 30 m and a depth of 10 m. The dimensions of the calculation domain, made nondimensional with the building height are depicted in Fig. 2. The numbers given in parentheses represent the nondimensional coordinates of the various corners of the calculation domain. For the turbulence model in the CFD calculation, the Reynolds Stress Model is used. Instead of assuming isotropic eddy viscosity, the RSM calculates the individual Reynolds stresses, using differential transport equations, together with an equation for the dissipation rate. The Reynolds Stress Model with the default parameters has superior performance for flows involving separation according to Shih, et al. 5, and the Fluent User s Guide 6. Close to the wall, the nonequilibrium wall function approach is used. The viscosity-affected region is not resolved, but instead bridged by the nonequilibrium wall functions Launder and Spalding 7 and Kim and Choudhury 8, since these functions are best suited for flows involving separation. In the nonequilibrium wall functions the Launder and Spalding s log law for the mean velocity is sensitive to pressure gradients. Furthermore, the twolayer-based concept is adopted to compute the budget of the turbulent kinetic energy in the wall-neighboring cells. By doing this, Table 1 Equivalent Surface Roughness Lengths Ground Cover Roughness length m Long grass or rocky ground 0.05 Pasture land 0.20 Suburban housing 0.6 Forests, cities 1 6 the nonequilibrium wall functions partly account for nonequilibrium effects that are neglected in the standard wall functions Fluent 5 user guide 6. The discretization is carried out with the first-order upwind scheme and the velocity-pressure flow field is determined using the SIMPLE model see Fluent 5 user guide 6. The nonstructured mesh is built of hybrid tetra grid cells, with the smallest cell size at the walls. The smallest cell dimension is equal to or larger than double the roughness height. Calculations with a second-order upwind scheme showed that numerical diffusion did not significantly influence the results of the first order upwind scheme. The input at the velocity inlet consists of the axial velocity u(z), the turbulent kinetic energy K and the turbulent dissipation rate (z) for a neutral boundary layer, all at a given height z. The boundary conditions at the inlet are based on the surface roughness in the domain so the boundary layer is unchanged in the entire domain. For a neutral boundary layer with ground surface roughness z 0, the logarithmic profile of the boundary layer reads uz u * ln zd z o. (1) In this, u is the friction-velocity, is the Von Karman constant * taken as 0.4, and d is the displacement length, which Panofsky and Dutton 3 claim is typically 75% of the average height of the roughness elements l: d0.75l. (2) The surface roughness z 0 is determined from measured wind profiles. Panofsky and Dutton 3 give the information in Table I for associating a roughness length with the description of the surface. According to Wilcox 9 the turbulence parameters at the inlet are: the turbulent kinetic energy K K u 2 * (3) C Fig. 1 Visualization of the turbulent separation over a rectangular building in boundary layer flow Fig. 2 Dimensions of the domain of the CFD calculation 434 Õ Vol. 125, NOVEMBER 2003 Transactions of the ASME

3 with k- model parameter C 0.09, and the turbulent dissipation (z) z u 3 * zd. (4) Negative or infinite values of (z) for zd are avoided by putting them equal to the value of (z) at the center of the first cell close the ground surface with (z)0. The CFD calculations yield the separation streamlines on the flat roof plotted in Fig. 3. From the height of the separation streamlines and the distance to the windward roof edge, the skew angle of the flow, as shown in Fig. 4, is calculated. The wiggling of the curve is caused by finite grid size. At the leading edge, the skew angle is equal to 90 deg, as a result of the deflection of the wind by the upwind wall. From Fig. 4, it can be concluded that the skew angle of the flow above the center of the flat roof of the building with a surrounding roughness of z m lies between 10 and 20 deg apart from close to the windward roof edge. For higher buildings the skew angle will increase, while for higher roughness the skew angle will decrease. The CFD calculation of the velocity above the center of the roof shows that the velocity gradient in the flow above the separation bubble (z27 m) is comparable with the gradient in the undisturbed flow see Fig. 5. It is common practice to treat this gradient as a second order effect for the performance of VAWTs in boundary layer flow since it can be showed see de Vries 10 that wind shear has a small effect on the power output of a VAWT. The same is assumed for the H-Darrieus in the flow above the roof. The H-Darrieus in Skewed Flow In the following, the single actuator disk multiple stream tube model first developed by Wilson in 11 will be used. The basic assumptions important for the model developed in this paper are that: the flow through the H-Darrieus can be divided into flow through multiple stream-tubes that do not interact with each other, the deceleration of the air by the rotor can be modeled with one actuator plate, the chord length of the blades c is much smaller than the radius R of the H-Darrieus. Schematically, the H-Darrieus can be represented by two rotor parts; one at the windward side and one at the leeward side of the H-Darrieus. In single actuator disk multiple streamtube theory, however, both parts are modelled with one actuator disk. For skewed flow with skew angle, the flow at the rotor can be split up as shown in Fig. 6 into two parts; one part of the flow will pass only one rotor part subscript s and one part of the flow subscript d will pass two rotor parts. In the following, the single rotor part variables will have a subscript s and the double rotor part will have a subscript d. The length of the parts depends on the skew angle of the flow and the induction factor see the following discussion on induction factor. For a high skew angle and a wide rotor, almost the whole H-Darrieus operates as two single rotor parts. According to the cross-flow principle mentioned in Jones and Cohen 12, the lift force on a blade in skewed flow is determined by the component perpendicular to the blade and is not dependent on the component parallel to the blade. This cross-flow principle is valid as long as skin friction is of no importance. The lift on a blade of the double rotor part in skewed flow can then be calculated based on the velocity component perpendicular to the blade V o,y,d, horizontal component which reads: V o,y,d V o cos a d, (5) where V o is the undisturbed velocity at the roof, is the skew angle and a d is the induction factor of the double rotor part. The Fig. 3 CFD calculations of the separation streamlines at a model building with dimensions heightãwidthãdepthä20ã30 Ã10 m for different roughness z 0 of the upwind area Fig. 4 CFD calculations of the skew angle deg at a model building with dimensions heightãwidthãdepthä20ã30ã10 m for different roughness z 0 of the upwind area Fig. 5 CFD calculation of the velocity profile above the center of the roof compared to the undisturbed velocity profile Journal of Solar Energy Engineering NOVEMBER 2003, Vol. 125 Õ 435

4 induced velocity, which is by definition set equal to a d V o, is assumed to be normal to the actuator disk see Fig. 7, since the force caused by the actuator plate is a pressure force and thus normal to the plate. According to Fig. 6 the flow through the double rotor part has an angle of: sin d arctan cos a d (6) with the normal vector at the actuator plate 2. The flow thus ceases to pass the rotor parts if the denominator is equal to zero. We therefore demand that: arccos a d. On the other hand, the horizontal component of the velocity downwind of the actuator disk (V o (cos 2a d )) equals zero or becomes negative for too high an induction factor. This causes the flow assumptions in the momentum theory to be violated. We therefore demand that: cos 2a d 0, which gives a d 1 cos 2 The lift L per unit blade length reads: 1 LV cd,d c l,d 2 V 2 cd,dc, (7) where is the air density, V cd,d is the resultant velocity on the blade see Fig. 8, is the strength of the bound vortex of the airfoil, c is the chord length of the blade, and c l,d is the lift coefficient of the airfoil of the double rotor part. The lift coefficient of the airfoil is approximated with a function that takes Reynolds effects and stall of the airfoil into account. The function is fit to the measurements of the NACA 0018 profile that can be found in Paraschivoiu 13. The force on the blade per unit blade length can be expressed as: F y,d 1 2 c l,dr sin cv cd,d, (9) with radius of the H-Darrieus R, rotational frequency, and rotational angle (0 corresponds to a blade moving parallel to the wind direction, see Fig. 8. The blade occupies a stream-tube width of: dxrdsin. (10) From Equation 9, it can be seen that the absolute value of the force parallel to the stream tube and with this, the local power production repeats itself after 360 deg. Therefore the analysis is limited to one revolution. For the double rotor part, the blade passes the flow in the streamtube twice. The blade thus stays in the streamtube for a fraction of 2d/2 of the total time of one revolution. So, Equation 9 should be multiplied with d/ to find the average blade force in the streamtube. For the single rotor part, this averaging factor is half this value since the blade passes the flow just one time. From Equation 9, the average force df bl,d for B blades in the direction of the streamtube at the double rotor part now follows as: d df bl,d Bc l,d R sin cv cd,d 2. (11) F d V cd,d. (8) With equation 7 this results in a blade force in the y-direction at the double rotor part of: 2 The first paper of the authors on the topic was published at the ASME Wind Energy Symposium 2003 contribution nr. AIAA In this paper, the change of the undisturbed flow angle by the induction factor was not taken into account. Fig. 7 disk Schematic drawing of the skewed flow on an actuator Fig. 6 Schematic side view of an H-Darrieus in skewed flow. Between the dotted lines: operating as a double rotor part d. Above and below the dotted lines: operating as a single rotor part s. Fig. 8 Top view of the H-Darrieus in skewed flow 436 Õ Vol. 125, NOVEMBER 2003 Transactions of the ASME

5 The blade force can also be found with momentum theory. Glauert momentum theory for yawed flow shows good agreement with measurements on a HAWT in yawed flow as shown by Madsen, Sorensen, and Schreck 14. It gives the thrust force perpendicular to the actuator plate, which is assumed to lie on the y-axis as shown in Burton et al. 15. The thrust force reads: df mo,d Rdsin W d 2a d V o (12) for the double rotor plate, where W d is the total resulting velocity towards the actuator plate see Fig. 7: W d V o sin 2 cos a d 2. (13) Equating 11 and 12 gives an implicit relation for the induction factor a d for B blades for a double rotor part in skewed flow: a d 1 Bc 4 R c l,d V cd,d. (14) W d where the Tip Speed Ratio is defined as: R. (15) V o The same derivation is carried out for the single rotor part where we have to keep in mind that the blade moves only one time through the flow. The result is: a s 1 Bc 8 R c l,s V cd,s. (16) W s Rotor Performance In the previous section, the flow angle of the double rotor part is found to be different from the single rotor part different induction factor. The flow through the double rotor part pushes aside the flow of the single rotor part see Fig. 9. It is assumed that the flows do not mix but instead have an average flow angle, which is the weighted average of the flow angles calculated from the induction factors. This average flow angle is denoted by. Fig. 9 Side view of the H-Darrieus with illustration of the different flow angles According to Fig. 6 and Fig. 8 the blade length for flow angles up to: max arctan H 2R (17) reads h s 2R tan sin (18) for the single rotor part and h d Hh s H2R tan sin (19) for the double rotor part. Larger skew angles cause the blade length of the double rotor part to disappear and the length of the blade for the single rotor part to be limited to h s H for: H H arcsin 2R tan arcsin 2R tan. (20) For outside these borders and max the blade length of the different rotor parts can again be found with Equations 18 and 19. The known blade lengths of the single and double rotor part enable us to derive an implicit relation for the flow angle, which is taken as a weighted average on the basis of those blade lengths. The flow angle is thus defined as: h s Hh s Hh s s Hh d. (21) s The extracted power dp per unit blade length in a stream tube with width dy can be found from the thrust force Equation 12 times the velocity through the rotor Equation 5: dpdf mo,skew V o,y, (22) since both the thrust force and the x-velocity point in the same direction. From the power the performance coefficient calculated on basis of the frontal area of the rotor reads: and dc P,d 2d sin W d a d h d H cos a d (23) dc P,s 2d sin W s a s h s H cos a s. (24) The viscous loss in performance for the double rotor part can be found from the drag coefficient of the blades c d,d : dc P,cd,d Bc d,d V cd,d V o and from c d,s for the single rotor part dc P,cd,s Bc d,s V cd,s V o 2 ch d d 2RH 2 (25) 2 ch s d 2RH 2. (26) In order to calculate the total performance coefficient of a rotor part dx within a stream tube we need to add the efficiencies and losses of the different parts as: dc P dc P,d 2dC P,s 2dC p,cd,d 2dC p,cd,s (27) where we counted the power production of the single rotor part twice since there is one single rotor part at the windward side of the rotor and one at the leeward side of the rotor. This fact is crucial for the operation of the H-Darrieus in skewed flow. It allows the power production to increase in skewed flow as a consequence of the increase in rotor area that experiences undisturbed flow. The maximum performance coefficient of the total rotor is the sum of the contributions of the single and double rotor part. For maximum performance coefficient of the separate parts, the single rotor part needs a higher TSR than the double rotor part. Journal of Solar Energy Engineering NOVEMBER 2003, Vol. 125 Õ 437

6 For maximum performance of the total rotor, the single rotor part will thus operate at a bit too low TSR and the double rotor part will operate at a bit too high TSR. However, the net result on the performance of both rotor parts, each operating below maximum performance, can be an increase of the turbine performance in skewed flow! The total performance coefficient, including drag losses, can be found from integration of dc P : Wind Tunnel Tests C P 0 dc P d. (28) In order to validate the analytical model of the H-Darrieus in skewed flow, tests were carried out in the open jet wind tunnel of the TU Delft Wind Energy Section. The diameter of the wind tunnel is 2.2 m and we tested a two bladed H-Darrieus of diameter m, height 0.5 m and chord length 0.08 m see Fig. 10 at wind tunnel velocity of 7 m/s. The airfoils of the Darrieus were NACA 0018 profiles. At maximum power for zero skew angle, the average induction factor at the double rotor area was calculated equations 23 to 27 to be 0.35 see the Results section, below. The tests consisted of measurement of the number of revolutions per minute under different loads for different skew angles. A Prony brake 3 was used to load the H-Darrieus. For the estimation of the maximum performance of the H-Darrieus, the load was increased with small steps until the rotor stopped turning. The performance at the highest load was the maximum. The net effect of the increased rotor area combined with the less than optimally loaded single and double rotor parts, as discussed after equation 27, thus causes the performance of the total rotor to increase. The calculations showed that the average induction factor at maximum performance for zero skew angle is Locally however, the induction factor is higher than 0.5 see Fig. 13. The assumed flow pattern in momentum theory is therefore violated zero velocity at the downwind part of the stream tube. For induction factors above approximately 0.4, one should use empirical evaluations see for instance Burton et al. 15 of the thrust force to overcome the breakdown of momentum theory. These relations, however, are not known for skew angles and are thus not integrated in the current model for the operation of an H-Darrieus in skewed flow. A first attempt to improve the model should therefore focus on the thrust force for heavily loaded H-Darrieus in skewed flow. Figure 12 shows that the predicted performance coefficient is very high compared to the measurements. Furthermore, there is a mismatch between data and prediction in the skew angle at which Results The measurements and calculations with the analytical model Equations 23 to 27 of the performance coefficient and Tip- Speed are presented in Figs. 11 and 12 as functions of the flow skew angle,. 3 A Prony brake is a disk on the power axis with a pulley around the circumference of the disk. One end of the pulley is fixed at a force sensor that measures the force in the rope and at the other end of the rope a load causes friction around the circumference of the disk. This friction force and thus the torque are known by the fact that the sum of the moments around the power axis is zero. Fig. 11 Measured and calculated Tip Speed Ratio at maximum aerodynamic efficiency in skewed flow with skew angle Fig. 10 Photo of the test of the H-Darrieus in the open-jet wind tunnel. The size of the Darrieus is overestimated because of the camera point of view Fig. 12 Measured and calculated performance coefficients in skewed flow with skew angle 438 Õ Vol. 125, NOVEMBER 2003 Transactions of the ASME

7 the performance coefficient and the Tip Speed Ratio start to decrease measured to be about 25 deg. The model performance coefficient prediction showed a better agreement with the measurements for a lift coefficient equal to 2 sin() maximum error in performance coefficient below 15%. Any discussion about possible causes of the mismatch between model and measurements, however, is speculative since the local induction factor sometimes violates the flow pattern assumed in momentum theory. The presented model should provide a first basis for an assessment of the performance of an H-Darrieus in skewed flow. Many modeling problems occur. One of the most important is undoubtedly the different flow angles through the single and double rotor parts. To what extent does the flow through the double rotor part move aside the flow through the single rotor part? The calculated difference in flow angle can be up to 30 deg at the heavily loaded center of the H-Darrieus. Besides focusing on the induction factor of heavily loaded Darrieus in skewed flow, future efforts to refine the model should thus focus on the problem of determining the flow angle Fig. 13 Calculated induction factor vertical at zero skew angle, plotted on the projected diameter of the H-Darrieus horizontal. Right part of the graph: blade moves against the wind Fig. 14 Maximum aerodynamic efficiency of a HAWT in yawed flow according to Glauert momentum theory for yawed flow 15 The Horizontal Axis Wind Turbine in Skewed Flow According to the Wind Energy Handbook 15, Glauert gives the thrust force of an actuator disk in yaw. This equation can be used to calculate the performance coefficient of a HAWT in skewed flow. C P 4a1a2 cos acos a. (29) According to Equation 29 the optimum performance coefficient of a HAWT thus shows a decrease for increasing angle as shown in Fig. 14. Conclusions The performance coefficient, C P, of an H-Darrieus in skewed flow, based on the projected frontal rotor area at zero skew angle, can increase above that for nonskewed flow. The effect has been measured and modeled and can be explained with the threedimensional shape of the H-Darrieus, a geometry that causes an effective increase of the energy extracting area in skewed flow. The maximum performance coefficient of a slender H-Darrieus (h/d1) in skewed flow increases less than a disk-like H-Darrieus (h/d1) because the relative increase in area experiencing undisturbed flow is less. The local induction factor in skewed flow can be well above the permitted induction factor of momentum theory. Therefore, momentum theory is no longer valid and the local performance predictions are questionable. The author has not found any useful research on the thrust force for high induction factor in skewed flow, so it appears that theory must be developed for this problem area. The angle of the flow through the disk is a function of the induction factors through the single and double rotor part. Since the induction factor of the double rotor part is the largest, the flow through the double rotor part moves towards the flow through the single rotor part, thus pushing aside the single-rotor flow. It is still unknown to what extent this interaction takes place. Changes in the model showed that this effect has a big influence on the performance. For a given dimension of the H-Darrieus, the Tip Speed Ratio at maximum performance increases in skewed flow. This increase must be taken into account when designing an H-Darrieus for skewed flow since the centrifugal force increases with the Tip Speed Ratio squared. On the Research The presented research is part of research into wind energy in the built environment, which will be the subject of a report to be published at the beginning of The work focuses on the aerodynamic aspects of wind turbines close to buildings. The authors thanks Simon Toet, Michiel Zaaijer and graduating student Joẗo Andre Monteiro Sardo of the wind energy group of Delft University of Technology for their indispensable assistance in the presented research. Nomenclature a induction factor of the H-Darrieus. C k- model turbulence model constant0.09. B number of blades of the H-Darrieus. c chord length of the airfoil blade m c l lift coefficient of the airfoil blade. C P performance coefficient of the H-Darrieus defined on the projected frontal area. c d drag coefficient of the airfoil. c P,cd performance coefficient loss caused by the drag of the airfoils. d displacement length m D diameter of the H-Darrieus m Journal of Solar Energy Engineering NOVEMBER 2003, Vol. 125 Õ 439

8 F bl The force parallel to the stream tube caused by the airfoil in the stream tube found with blade element theory per unit blade length N/m F mo The force parallel to the stream tube caused by the airfoil in the stream tube found with momentum theory per unit blade length N/m h height of the blade length of a rotor part double or single rotor part N/m H height of the H-Darrieus m K turbulent kinetic energy m 2 /s 2 l average height of the roughness elements m L lift force of the airfoil of the H-Darrieus per unit blade length N/m P power of the H-Darrieus per unit blade length Watt/ m R H-Darrieus radius m u undisturbed velocity in the boundary layer m/s u 0 undisturbed velocity at a certain height in the boundary layer m/s u friction velocity m/s * V o,y horizontal component of the total undisturbed velocity at the roof m/s V o total component of the undisturbed velocity at the roof m/s V cd resulting velocity at the airfoil blade perpendicular to the H-Darrieus axis m/s W total resulting velocity at a rotor part m/s x horizontal coordinate perpendicular to the direction of the undisturbed wind vector. y horizontal coordinate parallel to the direction of the undisturbed wind vector. z vertical coordinate perpendicular to the wind direction. z 0 roughness height of the surface m skew angle of the flow above the separation bubble on the roof. resulting flow angle through the rotor of the H-Darrieus. turbulent dissipation of kinetic energy. rotational angle of the H-Darrieus. Von Karman constant0.4. Tip Speed Ratio of the H-Darrieus. Strength of the bound vortex of the airfoil Subscripts d used for the double rotor part. s used for the single rotor part. References 1 Scheepers, M.J.J. et al., 2001, Energie Markt Trends, ECN report number: ECN-p Panofsky, H.A., Dutton, J.A., 1984, Atmospheric Turbulence, Wiley, New York. 4 Mertens, S., 2002, Wind Energy in Urban Areas, Refocus, the international renewable energy magazine. 5 Shih, T.-H., Liou, W.W., Shabbir, A., and Zhu, J., 1995, A New k-e Eddy- Viscosity Model for High Reynolds Number Turbulent Flows Model Development and Validation, Comput. Fluids, 243, pp Fluent, 1998, Fluent 5 User s Guide 2, Fluent Inc., Lebanon, USA, Launder, B.E., and Spalding, D.B., 1974, The Numerical Computation of Turbulent Flows, Comput. Methods Appl. Mech. Eng., 3, pp Kim, S.-E., and Choudhury, D., 1995, A Near-Wall Treatment Using Wall Functions Sensitized to Pressure Gradient, ASME FED Vol. 217, Separated and Complex Flows, ASME. 9 Wilcox, D.C., 1994, Turbulence Modeling for CFD, DCW Industries, Inc., pp de Vries, O., 1979, Fluid Dynamic Aspects of Wind Energy Conversion, AGARD AG 243, July. 11 Wilson, R.E., Lissaman, 1994, P.B.S., Applied Aerodynamics of Wind Power Machines, Aerovironment, Inc, Pasadena, CA. 12 Jones, R.T., Cohen, D., 1957, Aerodynamics of Wings at High Speeds, Aerodynamic Components of Aircraft at High Speeds, High Speed Aerodynamics and Jet Propulsion, Vol. VII, AF Donovan, HR Lawrence eds, Princeton University Press, Princeton, NJ. 13 Paraschivoiu, I., 2002, Wind Turbine Design With Emphasis on Darrieus Concept, Ecole polytechnique de Montreal, Madsen, H.A., Sorensen, N.N., Schreck, S., 2003, Yaw Aerodynamics Analyzed With Three Codes in Comparison With Experiment, ASME Wind Energy Symposium, contribution nr. AIAA Burton, T., Sharpe, D., Jenkins, N., Bossanyi, E., 2001, Wind Energy Handbook, J. Wiley & sons Ltd, England. 440 Õ Vol. 125, NOVEMBER 2003 Transactions of the ASME