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How wind Turbines produce Electricity

how wind turbines work and how wind turbines are installed and how wind turbines affect the environment and how wind turbine blades work
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3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 1 1 General Introduction to Wind Turbines Before addressing more technical aspects of wind turbine technology, an attempt is made to give a short general introduction to wind energy. This involves a very brief historical part explaining the development of wind power, as well as a part dealing with economy and wind turbine design. It is by no means the intention to give a full historical review of wind turbines, merely to mention some major milestones in their development and to give examples of the historical exploitation of wind power. Short Historical Review The force of the wind can be very strong, as can be seen after the passage of a hurricane or a typhoon. Historically, people have harnessed this force peacefully, its most important usage probably being the propulsion of ships using sails before the invention of the steam engine and the internal combust- ion engine. Wind has also been used in windmills to grind grain or to pump water for irrigation or, as in The Netherlands, to prevent the ocean from flooding low-lying land. At the beginning of the twentieth century electricity came into use and windmills gradually became wind turbines as the rotor was connected to an electric generator. The first electrical grids consisted of low-voltage DC cables with high losses. Electricity therefore had to be generated close to the site of use. On farms, small wind turbines were ideal for this purpose and in Denmark Poul la Cour, who was among the first to connect a windmill to a generator, gave a course for ‘agricultural electricians’. An example of La Cour’s great foresight was that he installed in his school one of the first wind tunnels in the world in order to investigate rotor aerodynamics. Gradually, however, diesel engines and steam turbines took over the production of electricity and only during the two world wars, when the supply of fuel was scarce, did wind power flourish again.3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 2 2 Aerodynamics of Wind Turbines However, even after the Second World War, the development of more efficient wind turbines was still pursued in several countries such as Germany, the US, France, the UK and Denmark. In Denmark, this work was undertaken by Johannes Juul, who was an employee in the utility company SEAS and a former student of la Cour. In the mid 1950s Juul introduced what was later called the Danish concept by constructing the famous Gedser turbine, which had an upwind three-bladed, stall regulated rotor, connected to an AC asynchronous generator running with almost constant speed. With the oil crisis in 1973, wind turbines suddenly became interesting again for many countries that wanted to be less dependent on oil imports; many national research programmes were initiated to investigate the possibilities of utilizing wind energy. Large non-commercial prototypes were built to evaluate the economics of wind produced electricity and to measure the loads on big wind turbines. Since the oil crisis, commercial wind turbines have gradually become an important industry with an annual turnover in the 1990s of more than a billion US dollars per year. Since then this figure has increased by approximately 20 per cent a year. Why Use Wind Power? As already mentioned, a country or region where energy production is based on imported coal or oil will become more self-sufficient by using alternatives such as wind power. Electricity produced from the wind produces no CO 2 emissions and therefore does not contribute to the greenhouse effect. Wind energy is relatively labour intensive and thus creates many jobs. In remote areas or areas with a weak grid, wind energy can be used for charging batteries or can be combined with a diesel engine to save fuel whenever wind is available. Moreover, wind turbines can be used for the desalination of water in coastal areas with little fresh water, for instance the Middle East. At windy sites the price of electricity, measured in /kWh, is competitive with the production price from more conventional methods, for example coal fired power plants. To reduce the price further and to make wind energy more competitive with other production methods, wind turbine manufacturers are concen- trating on bringing down the price of the turbines themselves. Other factors, such as interest rates, the cost of land and, not least, the amount of wind available at a certain site, also influence the production price of the electrical energy generated. The production price is computed as the investment plus the discounted maintenance cost divided by the discounted production measured in kWh over a period of typically 20 years. When the character-3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 3 General Introduction to Wind Turbines3 istics of a given turbine – the power for a given wind speed, as well as the annual wind distribution – are known, the annual energy production can be estimated at a specific site. Some of the drawbacks of wind energy are also mentioned. Wind turbines create a certain amount of noise when they produce electricity. In modern wind turbines, manufacturers have managed to reduce almost all mechanical noise and are now working hard on reducing aerodynamic noise from the rotating blades. Noise is an important competition factor, especially in densely populated areas. Some people think that wind turbines are unsightly in the landscape, but as bigger and bigger machines gradually replace the older smaller machines, the actual number of wind turbines will be reduced while still increasing capacity. If many turbines are to be erected in a region, it is important to have public acceptance. This can be achieved by allowing those people living close to the turbines to own a part of the project and thus share the income. Furthermore, noise and visual impact will in the future be less important as more wind turbines will be sited offshore. One problem is that wind energy can only be produced when nature supplies sufficient wind. This is not a problem for most countries, which are connected to big grids and can therefore buy electricity from the grid in the absence of wind. It is, however, an advantage to know in advance what resources will be available in the near future so that conventional power plants can adapt their production. Reliable weather forecasts are desirable since it takes some time for a coal fired power plant to change its product- ion. Combining wind energy with hydropower would be perfect, since it takes almost no time to open or close a valve at the inlet to a water turbine and water can be stored in the reservoirs when the wind is sufficiently strong. The Wind Resource A wind turbine transforms the kinetic energy in the wind to mechanical energy in a shaft and finally into electrical energy in a generator. The maxi- mum available energy, P , is thus obtained if theoretically the wind speed max · 2 3 · could be reduced to zero: P = 1/2 mV = 1/2 AV where m is the mass flow, o o V is the wind speed,  the density of the air and A the area where the wind o speed has been reduced. The equation for the maximum available power is very important since it tells us that power increases with the cube of the wind speed and only linearly with density and area. The available wind speed at a given site is therefore often first measured over a period of time before a project is initiated.3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 4 4 Aerodynamics of Wind Turbines In practice one cannot reduce the wind speed to zero, so a power coefficient C is defined as the ratio between the actual power obtained and p the maximum available power as given by the above equation. A theoretical maximum for C exists, denoted by the Betz limit, C = 16/27 = 0.593. p Pmax Modern wind turbines operate close to this limit, with C up to 0.5, and are p therefore optimized. Statistics have been given on many different turbines sited in Denmark and as rule of thumb they produce approximately 2 1000kWh/m /year. However, the production is very site dependent and the rule of thumb can only be used as a crude estimation and only for a site in Denmark. Sailors discovered very early on that it is more efficient to use the lift force than simple drag as the main source of propulsion. Lift and drag are the components of the force perpendicular and parallel to the direction of the relative wind respectively. It is easy to show theoretically that it is much more efficient to use lift rather than drag when extracting power from the wind. All modern wind turbines therefore consist of a number of rotating blades looking like propeller blades. If the blades are connected to a vertical shaft, the turbine is called a vertical-axis machine, VAWT, and if the shaft is horizontal, the turbine is called a horizontal-axis wind turbine, HAWT. For commercial wind turbines the mainstream mostly consists of HAWTs; the following text therefore focuses on this type of machine. A HAWT as sketched in Figure 1.1 is described in terms of the rotor diameter, the number of blades, the tower height, the rated power and the control strategy. D H Figure 1.1 Horizontal-axis wind turbine (HAWT)3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 5 General Introduction to Wind Turbines5 The tower height is important since wind speed increases with height above the ground and the rotor diameter is important since this gives the area A in the formula for the available power. The ratio between the rotor diameter D and the hub height H is often approximately one. The rated power is the maximum power allowed for the installed generator and the control system must ensure that this power is not exceeded in high winds. The number of blades is usually two or three. Two-bladed wind turbines are cheaper since they have one blade fewer, but they rotate faster and appear more flickering to the eyes, whereas three-bladed wind turbines seem calmer and therefore less disturbing in a landscape. The aerodynamic efficiency is lower on a two- bladed than on a three-bladed wind turbine. A two-bladed wind turbine is often, but not always, a downwind machine; in other words the rotor is downwind of the tower. Furthermore, the connection to the shaft is flexible, the rotor being mounted on the shaft through a hinge. This is called a teeter mechanism and the effect is that no bending moments are transferred from the rotor to the mechanical shaft. Such a construction is more flexible than the stiff three-bladed rotor and some components can be built lighter and smaller, which thus reduces the price of the wind turbine. The stability of the more flexible rotor must, however, be ensured. Downwind turbines are noisier than upstream turbines, since the once-per-revolution tower passage of each blade is heard as a low frequency noise. The rotational speed of a wind turbine rotor is approximately 20 to 50 rpm and the rotational speed of most generator shafts is approximately 1000 to 3000 rpm. Therefore a gearbox must be placed between the low-speed rotor shaft and the high-speed generator shaft. The layout of a typical wind turbine can be seen in Figure 1.2, showing a Siemens wind turbine designed for offshore use. The main shaft has two bearings to facilitate a possible replace- ment of the gearbox. This layout is by no means the only option; for example, some turbines are equipped with multipole generators, which rotate so slowly that no gearbox is needed. Ideally a wind turbine rotor should always be perpendicular to the wind. On most wind turbines a wind vane is therefore mounted somewhere on the turbine to measure the direction of the wind. This signal is coupled with a yaw motor, which continuously turns the nacelle into the wind. The rotor is the wind turbine component that has undergone the greatest development in recent years. The aerofoils used on the first modern wind turbine blades were developed for aircraft and were not optimized for the much higher angles of attack frequently employed by a wind turbine blade. Even though old aerofoils, for instance NACA63-4XX, have been used in the light of experience gained from the first blades, blade manufacturers have3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 6 6 Aerodynamics of Wind Turbines With permission from Siemens Wind Power. Figure 1.2 Machine layout now started to use aerofoils specifically optimized for wind turbines. Different materials have been tried in the construction of the blades, which must be sufficiently strong and stiff, have a high fatigue endurance limit, and be as cheap as possible. Today most blades are built of glass fibre reinforced plastic, but other materials such as laminated wood are also used. It is hoped that the historical review, the arguments for supporting wind power and the short description of the technology set out in this chapter will motivate the reader to study the more technical sections concerned with aerodynamics, structures and loads as applied to wind turbine construction.3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 7 2 2-D Aerodynamics Wind turbine blades are long and slender structures where the spanwise velocity component is much lower than the streamwise component, and it is therefore assumed in many aerodynamic models that the flow at a given radial position is two dimensional and that 2-D aerofoil data can thus be applied. Two-dimensional flow is comprised of a plane and if this plane is described with a coordinate system as shown in Figure 2.1, the velocity component in the z-direction is zero. In order to realize a 2-D flow it is necessary to extrude an aerofoil into a wing of infinite span. On a real wing the chord and twist changes along the span and the wing starts at a hub and ends in a tip, but for long slender wings, like those on modern gliders and wind turbines, Prandtl has shown that local 2-D data for the forces can be used if the angle of attack is corrected accordingly with the trailing vortices behind the wing (see, for example, Prandtl and Tietjens, 1957). These effects will be dealt with later, but it is now clear that 2-D aerodynamics is of practical interest even though it is difficult to realize. Figure 2.1 shows the leading edge stagnation point present in the 2-D flow past an aerofoil. The reacting force F from the flow is decomposed into a direction perpendicular to the velocity at infinity V and ∝ to a direction parallel to V . The former component is known as the lift, L; ∝ the latter is called the drag, D (see Figure 2.2). y z x Figure 2.1 Schematic view of streamlines past an airfoil3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 8 8 Aerodynamics of Wind Turbines Figure 2.2 Definition of lift and drag If the aerofoil is designed for an aircraft it is obvious that the L/D ratio should be maximized. The lift is the force used to overcome gravity and the higher the lift the higher the mass that can be lifted off the ground. In order to maintain a constant speed the drag must be balanced by a propulsion force delivered from an engine, and the smaller the drag the smaller the required engine. Lift and drag coefficients C and C are defined as: l d L C = ––––––––– (2.1) l 2 1 / ρV c 2 ∝ and: D C = ––––––––– (2.2) d 2 1 / ρV c 2 ∝ where ρ is the density and c the length of the aerofoil, often denoted by the chord. Note that the unit for the lift and drag in equations (2.1) and (2.2) is force per length (in N/m). A chord line can be defined as the line from the trailing edge to the nose of the aerofoil (see Figure 2.2). To describe the forces completely it is also necessary to know the moment M about a point in the aerofoil. This point is often located on the chord line at c/4 from the3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 9 2–D Aerodynamics9 Figure 2.3 Explanation of the generation of lift leading edge. The moment is positive when it tends to turn the aerofoil in Figure 2.2 clockwise (nose up) and a moment coefficient is defined as: M C = –––––––––– (2.3) m 2 2 1 / ρV c 2 ∝ The physical explanation of the lift is that the shape of the aerofoil forces the streamlines to curve around the geometry, as indicated in Figure 2.3. From basic fluid mechanics it is known that a pressure gradient, ∂p/∂r = 2 V /r, is necessary to curve the streamlines; r is the curvature of the stream- line and V the speed. This pressure gradient acts like the centripetal force known from the circular motion of a particle. Since there is atmospheric pressure p far from the aerofoil there must thus be a lower than atmospheric o pressure on the upper side of the aerofoil and a higher than atmospheric pressure on the lower side of the aerofoil. This pressure difference gives a lifting force on the aerofoil. When the aerofoil is almost aligned with the3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 10 10 Aerodynamics of Wind Turbines flow, the boundary layer stays attached and the associated drag is mainly caused by friction with the air. The coefficients C , C and C are functions of α, Re and Ma. α is the angle l d m of attack defined as the angle between the chordline and V; Re is the ∝ Reynolds number based on the chord and V, Re = cV /ν, where ν is the ∝ ∝ kinematic viscosity; and Ma denotes the Mach number, in other words the ratio between V and the speed of sound. For a wind turbine and a slow ∝ moving aircraft the lift, drag and moment coefficients are only functions of α and Re. For a given airfoil the behaviours of C , C and C are measured l d m or computed and plotted in so-called polars. An example of a measured polar for the FX67-K-170 airfoil is shown in Figure 2.4. C increases linearly with , with an approximate slope of 2/rad, until a l certain value of α, where a maximum value of C is reached. Hereafter the l aerofoil is said to stall and C decreases in a very geometrically dependent l manner. For small angles of attack the drag coefficient C is almost constant, d but increases rapidly after stall. The Reynolds number dependency can also be seen in Figure 2.4. It is seen, especially on the drag, that as the Reynolds number reaches a certain value, the Reynolds number dependency becomes small. The Reynolds number dependency is related to the point on the aerofoil, where the boundary layer transition from laminar to turbulent flow occurs. The way an aerofoil stalls is very dependent on the geometry. Thin aerofoils with a sharp nose, in other words with high curvature around the leading edge, tend to stall more abruptly than thick aerofoils. Different stall behaviours are seen in Figure 2.5, where C (α) is compared for two different l aerofoils. The FX38-153 is seen to lose its lift more rapidly than the FX67- K-170. The explanation lies in the way the boundary layer separates from the upper side of the aerofoil. If the separation starts at the trailing edge of the aerofoil and increases slowly with increasing angle of attack, a soft stall is observed, but if the separation starts at the leading edge of the aerofoil, the entire boundary layer may separate almost simultaneously with a dramatic loss of lift. The behaviour of the viscous boundary layer is very complex and depends, among other things, on the curvature of the aerofoil, the Reynolds number, the surface roughness and, for high speeds, also on the Mach number. Some description of the viscous boundary is given in this text but for a more elaborate description see standard textbooks on viscous boundary layers such as White (1991) and Schlichting (1968). Figure 2.6 shows the computed streamlines for a NACA63-415 aerofoil at angles of attack of 5° and 15°. For α = 15° a trailing edge separation is observed. The forces on the aerofoil stem from the pressure distribution p(x)3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 11 2–D Aerodynamics11 Figure 2.4 Polar for the FX67-K-170 airfoil3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 12 12 Aerodynamics of Wind Turbines Figure 2.5 Different stall behaviour Figure 2.6 Computed streamlines for angles of attack of 5° and 15°3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 13 2–D Aerodynamics13 and the skin friction with the air  = µ (∂u/∂y) . (x,y) is the surface w y=0 coordinate system as shown in Figure 2.7 and µ is the dynamic viscosity. The skin friction is mainly contributing to the drag, whereas the force found from integrating the pressure has a lift and drag component. The drag component from the pressure distribution is known as the form drag and becomes very large when the aerofoil stalls. The stall phenomenon is closely related to separation of the boundary layer (see next paragraph); therefore rule number one in reducing drag is to avoid separation. In Abbot and von Doenhoff (1959) a lot of data can be found for the National Advisory Committee for Aeronautics (NACA) aerofoils, which have been extensively used on small aircraft, wind turbines and helicopters. Close to the aerofoil there exists a viscous boundary layer due to the no-slip condition on the velocity at the wall (see Figure 2.7). A boundary layer thickness is often defined as the normal distance δ(x) from the wall where u(x)/U(x) = 0.99. Further, the displacement thickness δ (x), the momentum thickness θ(x) and the shape factor H(x) are defined as: δ u δ (x) = (1– —)dy, (2.4) ∫ 0 U δ u u θ(x)= — (1– —)dy, and (2.5) ∫ 0 U U δ . H(x)= – (2.6) θ The coordinate system (x,y) is a local system, where x = 0 is at the leading edge stagnation point and y is the normal distance from the wall. A turbulent boundary layer separates for H between 2 and 3. The stagnation streamline (see Figure 2.1) divides the fluid that flows over the aerofoil from the fluid that flows under the aerofoil. At the stagnation point the velocity is zero and the boundary layer thickness is small. The fluid which flows over the aerofoil accelerates as it passes the leading edge and, since the leading edge is close to the stagnation point and the flow accelerates, the boundary layer is thin. It is known from viscous boundary layer theory (see, for example, White, 1991) that the pressure is approximately constant from the surface to the edge of the boundary layer, i.e. ∂p/∂y = 0. Outside the boundary layer the Bernoulli equation (see Appendix A) is valid and, since the flow accelerates, the pressure decreases, i.e. ∂p/∂x0. On the lower side the pressure gradient is much smaller since the curvature of the wall is small compared to the leading edge. At the trailing edge the pressure must be the same at the upper and lower side (the Kutta condition) and therefore the pressure must rise, ∂p/∂x0, from a minimum value somewhere on the upper side to a higher3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 14 14 Aerodynamics of Wind Turbines Figure 2.7 Viscous boundary layer at the wall of an airfoil Figure 2.8 Schematic view of the shape of the boundary layer for a favourable and an adverse pressure gradient value at the trailing edge. An adverse pressure gradient, ∂p/∂x0, may lead to separation. This can be seen directly from the Navier-Stokes equations (see Appendix A) which applied at the wall, where the velocity is zero, reduces to: 2 ∂ u 1 ∂p –— = – — (2.7) 2 ∂y ∂x The curvature of the u-velocity component at the wall is therefore given by the sign of the pressure gradient. Further, it is known that ∂u/∂y = 0 at y = δ.3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 15 2–D Aerodynamics15 From this can be deduced that the u velocity profile in an adverse pressure gradient, ∂p/∂x 0, is S-shaped and separation may occur, whereas the curv- ature of the u velocity profile for ∂p/∂x 0 is negative throughout the entire boundary layer and the boundary layer stays attached. A schematic picture showing the different shapes of the boundary layer is given in Figure 2.8. Since the form drag increases dramatically when the boundary layer separ- ates, it is of utmost importance to the performance of an aerofoil to control the pressure gradient. For small values of x the flow is laminar, but for a certain x the laminar trans boundary layer becomes unstable and a transition from laminar to turbulent flow occurs. At x the flow is fully turbulent. In Figure 2.9 transition from a T laminar to a turbulent boundary layer is sketched. The transitional process is very complex and not yet fully understood, but a description of the phenomena is found in White (1991), where some engineering tools to predict x are also given. One of the models which is sometimes used in trans aerofoil computations is called the one-step method of Michel. The method predicts transition when: 0.4 Re = 2.9 Re (2.8) θ x where Re = U(x)⋅θ(x)/ν and Re = U(x)⋅ x/ν. For a laminar aerofoil (see later), θ x however, the Michel method might be inadequate and more advanced 9 methods such as the e method (see White, 1991) should be applied. Figure 2.9 Schematic view of the transitional process Turbulent flow is characterized by being more stable in regions of adverse pressure gradients, ∂p/∂x0, and by a steeper velocity gradient at the wall, ∂u/∂y . The first property is good since it delays stall, but the second y = 0 property increases the skin friction and thus the drag. These two phenomena3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 16 16 Aerodynamics of Wind Turbines are exploited in the design of high performance aerofoils called laminar aerofoils. A laminar aerofoil is an aerofoil where a large fraction of the boundary layer is laminar and attached in the range it is designed for. To design such an aerofoil it is necessary to specify the maximum angle of attack, where the boundary layer to a large extent is supposed to be laminar. The aerofoil is then constructed so that the velocity at the edge of the boundary layer, U(x), is constant after the acceleration past the leading edge and downstream. It is known from boundary layer theory (see White, 1991, and Schlichting, 1968) that the pressure gradient is expressed by the velocity outside the boundary layer as: dp dU(x) –– = –U(x) ––––– (2.9) dx dx At this angle the pressure gradient is therefore zero and no separation will occur. For smaller angles of attack the flow U(x) will accelerate and dp/dx becomes negative, which again avoids separation and is stabilizing for the laminar boundary layer, thus delaying transition. At some point x at the upper side of the aerofoil it is, however, necessary to decelerate the flow in order to fulfil the Kutta condition; in other words the pressure has to be unique at the trailing edge. If this deceleration is started at a position where the boundary layer is laminar, the boundary layer is likely to separate. Just after the laminar/ turbulent transition the boundary layer is relatively thin and the momentum close to the wall is relatively large and is therefore capable of withstanding a high positive pressure gradient without separation. During the continuous deceleration towards the trailing edge the ability of the boundary layer to withstand the positive pressure gradient diminishes, and to avoid separation it is therefore necessary to decrease the deceleration towards the trailing edge. It is of utmost importance to ensure that the boundary layer is turbulent before decelerating U(x). To ensure this, a turbulent transition can be triggered by placing a tripwire or tape before the point of deceleration. A laminar aerofoil is thus characterized by a high value of the lift to drag ratio C /C below the design l d angle. But before choosing such an aerofoil it is important to consider the stall characteristic and the roughness sensitivity. On an aeroplane it is necessary to fly with a high C at landing since the speed is relatively small. If the pilot l exceeds C and the aerofoil stalls, it could be disastrous if C drops as l ,max l drastically with the angle of attack as on the FX38-153 in Figure 2.5. The aeroplane would then lose its lift and might slam into the ground. If the aerofoil is sensitive to roughness, good performance is lost if the wings are contaminated by dust, rain particles or insects, for example. On a wind turbine this could alter the performance with time if, for instance, the3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 17 2–D Aerodynamics17 turbine is sited in an area with many insects. If a wind turbine is situated near the coast, salt might build up on the blades if the wind comes from the sea, and if the aerofoils used are sensitive to roughness, the power output from the turbine will become dependent on the direction of the wind. Fuglsang and Bak (2003) describe some attempts to design aerofoils specifically for use on wind turbines, where insensitivity to roughness is one of the design targets. To compute the power output from a wind turbine it is necessary to have data of C (α,Re) and C (α,Re) for the aerofoils applied along the blades. l d These data can be measured or computed using advanced numerical tools, but since the flow becomes unsteady and three-dimensional after stall, it is difficult to obtain reliable data for high angles of attack. On a wind turbine very high angles of attack may exist locally, so it is often necessary to extrapolate the available data to high angles of attack. References Abbot, H. and von Doenhoff, A. E. (1959) Theory of Wing Sections, Dover Publications, New York Fuglsang, P. and Bak, C. (2003) ‘Status of the Risø wind turbine aerofoils’, presented at the European Wind Energy Conference, EWEA, Madrid, 16–19 June Prandtl, L. and Tietjens, O. G. (1957) Applied Hydro and Aeromechanics, Dover Publications, New York Schlichting, H. (1968) Boundary-Layer Theory, McGraw-Hill, New York White, F. M. (1991) Viscous Fluid Flow, McGraw-Hill, New York3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 18 3 3-D Aerodynamics This chapter describes qualitatively the flow past a 3-D wing and how the spanwise lift distribution changes the upstream flow and thus the local angle of attack. Basic vortex theory, as described in various textbooks (for example Milne-Thomsen, 1952), is used. Since this theory is not directly used in the Blade Element Momentum method derived later, it is only touched on very briefly here. This chapter may therefore be quite abstract for the reader with limited knowledge of vortex theory, but hopefully some of the basic results will be qualitatively understood. A wing is a beam of finite length with aerofoils as cross-sections and therefore a pressure difference between the lower and upper sides is created, giving rise to lift. At the tips are leakages, where air flows around the tips from the lower side to the upper side. The streamlines flowing over the wing will thus be deflected inwards and the streamlines flowing under the wing The wing is seen from the suction side. The streamline flowing over the suction side (full line) is deflected inwards and the streamline flowing under (dashed line) is deflected outwards. Figure 3.1 Streamlines flowing over and under a wing3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 19 3–D Aerodynamics19 will be deflected outwards. Therefore at the trailing edge there is a jump in the tangential velocity (see Figures 3.1 and 3.2). A jump in the tangential velocity is seen, due to the leakage at the tips. Figure 3.2 Velocity vectors seen from behind a wing Because of this jump there is a continuous sheet of streamwise vorticity in the wake behind a wing. This sheet is known as the trailing vortices. In classic literature on theoretical aerodynamics (see, for example, Milne- Thomsen, 1952), it is shown that a vortex filament of strength Γ can model the flow past an aerofoil for small angles of attack. This is because the flow for small angles of attack is mainly inviscid and governed by the linear Laplace equation. It can be shown analytically that for this case the lift is given by the Kutta-Joukowski equation: L = ρV Γ. (3.1) ∝ An aerofoil may be thus substituted by one vortex filament of strength Γ and the lift produced by a 3-D wing can be modelled for small angles of attack by a series of vortex filaments oriented in the spanwise direction of the wing, known as the bound vortices. According to the Helmholtz theorem (Milne- Thomsen, 1952), a vortex filament, however, cannot terminate in the interior of the fluid but must either terminate on the boundary or be closed. A complete wing may be modelled by a series of vortex filaments, Γ, i = i 1,2,3,4,..., which are oriented as shown in Figure 3.3. In a real flow the trailing vortices will curl up around the strong tip vortices and the vortex system will look more like that in Figure 3.4.3212 J&J Aerodynamic Turbines 15/11/07 1:42 PM Page 20 20 Aerodynamics of Wind Turbines Figure 3.3 Asimplified model of the vortex system on a wing The model based on discrete vortices, as shown in Figure 3.3, is called the lifting line theory (see Schlichting and Truckenbrodt, 1959 for a complete description). The vortices on the wing (bound vortices) model the lift, and the trailing vortices (free vortices) model the vortex sheet stemming from the three dimensionality of the wing. The free vortices induce by the Biot-Savart law a downwards velocity component at any spanwise position of the wing. For one vortex filament of strength Γ the induced velocity at a point p is (see Figure 3.5): Γ r ds   ∫ w =— ——— . (3.2) 3 4 r Figure 3.4 More realistic vortex system on a wing