Aerofoils, Lift, Drag and Circulation

Transcription

1 Aerofoils, Lift, Drag and Circulation 4.0 Introduction In Section 3.3 we saw that the assumption that blade elements behave as aerofoils allows the analysis of wind turbines in terms of the lift and drag coefficients, c l and c d respectively, of the aerofoil sections that comprise each blade. For a great many aerofoils, these coefficients are known from wind tunnel investigations. In this Chapter we study the basic features of aerofoils and emphasise aspects of their performance that are important for wind turbine application. It is important to remember that an aerofoil is two-dimensional and that, as indicated in Chapter 3, there are always assumptions involved in applying two-dimensional data to rotating, complex threedimensional geometries such as wind turbine blades. 4.1 Geometry and Definition of Aerofoils Figure 4.1 shows four members of the NACA four digit series. NACA the U.S. National Advisory Committee on Aeronautics was the forerunner to NASA, and was very active in aerofoil development after the First World War. The four digit family of aerofoils was developed in the late 1920s, and has the advantage of being analytically described, whereas most modern sections are specified by their surface co-ordinates. We will use the four digit family to illustrate the basic terminology of aerofoils which sprang from the manually-intensive methods of wing and propeller blade layout used in the early days of the aircraft industry. N A C A N A C A N A C A N A C A Figure 4.1: Several NACA four digit aerofoils of the same thickness and increasing camber. The dashed line is the mean line and the chord line is dotted. The most forward point on the aerofoil is the leading edge, which is on the left for the four aerofoils in Figure 4.1; the airflow is from left to right and the lift, l, acts upwards. Virtually all aerofoils have a sharp trailing edge, at the right hand end of the section in Figure 4.1. The straight, dotted line joining the leading and trailing edges is the chord line whose length, c, we have met previously. The angle of attack, α, which we have also met previously, is defined as the angle between the free stream velocity, U 0, and the chord line. If U 0 were in the horizontal direction, a positive α would occur by

2 Figure 4.2 The SG series of aerofoils for small wind turbines. See Table 4.1 for thickness and camber raising the leading edges. The chord line was normally the first quantity to be laid out for an aircraft wing. Then the mean line, shown dotted in Figure 4.1, is added. It lies mid-way (the mean position) between the upper and lower surfaces of an aerofoil. The maximum distance between the chord line and the mean line is called the camber. Finally, the aerofoil thickness is added. The NACA four digit aerofoils are no longer used extensively, but the 0012 is probably the most studied aerofoil in history see McCroskey (1987). As indicated by the coincidence of the mean line and chord, this aerofoil is symmetric, so that it produces no lift at zero angle of attack. This is a very useful property for aircraft tail fins, as well as helicopter and other blades. The 12 in the designation indicates that this aerofoil, as well as the other three shown, has a maximum thickness, t, of 12% of the chord, although some sections specifically designed for the root areas of wind turbine blades, are thicker to accommodate the large centrifugal and other stresses in this region. The first of the four numbers indicates the camber as a percentage of c; this must be zero for a symmetric section. The last aerofoil, the 6412, has 6% camber; this is about the upper limit for practical aerofoils. The second number in the aerofoil designation, 4 in every case, indicates that the camber occurs at 40% of c. This is the position of maximum thickness for all members for the NACA four digit family. Examples of the more modern SG aerofoils are shown in Figure 4.2. These were very recently designed by Professor Michael Selig (S) and Phillipe Giguerre (G) of the University of Illinios at Urbana-Champaign, for small wind turbines, and are probably the only aerofoils designed specifically for that purpose. The basic geometric parameters of these sections and their design parameters are given in Table 4.1 taken from Giguerre & Selig (1998). These authors consider other aerofoil sections for small turbines in Giguerre & Selig (1997). The 16% thick SG6040 is a root aerofoil, while the other three, intended for the power-extracting outer parts of the blade, have 10% thickness with varying camber. Some of the important features of the design methodology will be discussed in the next Section. 4.2 Aerofoil Performance Aerofoil t/c (%) camber (%) Design c l Design Re SG ,000 SG ,000 SG ,333 SG ,000 Table 4.1 Geometric parameters of the SG family An aerofoil of a given shape will have a lift, l, and drag, d, dependent on U 0 (which for aerofoils is equivalent to the effective velocity U T for blade elements), c, ρ, ν, and the angle of attack α. This leads to the definition of the lift and drag coefficients as: Aerofoils, Lift, Drag and Circulation 2

3 Figure 4.3: Lift coefficient for SG6040 aerofoil. Figure 4.4: Drag coefficient for SG6040 aerofoil. Figure 4.5: Drag polars for SG6040 aerofoil. Figure 4.6: Drag polars for SG6040 aerofoil. 3 Aerofoils, Lift, Drag and Circulation

4 c l = 1 2 l ρu 2 0 c d and cd = 1 2 ρu c 2 0 (4.1) and the basic formulation: c l = c ( α, Re) and c = l d c d ( α, Re) (4.2) where the Reynolds number, Re = U 0 c/ν, is directly analogous to the blade element definition in Equation (1.8). Note very carefully that the lift and drag that appear in (4.1) have the units of Newtons/m (N/m). This is because an aerofoil is two-dimensional and it makes no sense to think of the lift or drag depending on its length out of the page for Figures 4.1 and 4.2. One consequence of this is the appearance of the radial extent of the blade element, dr, in Equations (3.9) and (3.10) for the thrust and torque in terms of c l and c d. We now look at how the lift and drag depend on α and Re. Figures 4.3 and 4.4 give this information for the SG6040 aerofoil and we begin by considering the results for Re = 500,000 first. There is a range of α for which c l is nearly linear; there is a branch of theoretical aerodynamics called thin aerofoil theory which predicts this linear dependence. However, the linearity ends before 10, where c d in Figure 4.4 begins to increase rapidly; c d is almost constant over much of the linear range in c l. Between α = 10 and 15, the flow separates from the upper surface which stalls the aerofoil. The so-called drag polar is shown in Figure 4.5 and l/d in Figure 4.6. Notice that the maximum l/d occurs before the end of the linear region and recall from the previous Chapter that it is the ratio that is important in wind turbine design, rather than the individual values of lift and drag. Figures 4.3 and 4.4 show a gradual increase in c l and a decrease in c d with increasing Re. This trend continues above Re = 500,000. In a review of the large number of wind tunnel measurements of the NACA 0012 section, McCroskey (1987), suggested the following empirical formulae for the linear part of the lift curve: dc l /dα = log 10 (Re/10 6 ) (4.3) and for the minimum drag coefficient, c d0 : c d0 = Re (4.4) for Re 500,000 approximately. Thus the lift increases and drag decreases as Re increases. It is generally the case, however, that the changes are not great in regions where relations like (4.3) and (4.4) apply. Equations (4.3) and (4.4) are used in the program described in Chapter 3. From the answers to the Exercises in Chapter 1, it can be inferred that most large wind turbine blades operate at Re > 500,000, at least near the tip, where, as we will see in Chapter 5, most power is produced. However, this Re is the maximum reached on the 5 kw turbine described in that Chapter, while the minimum operating Re of the 600 W blades was only 6,000. Thus small wind turbines nearly always operate at low Reynolds number and we will look further at the alteration in aerofoil performance as Reynolds number decreases. Before looking at the effects of low Reynolds number, it is important to understand the relationship between the lift and the pressure distribution around an aerofoil. Figure 4.7 shows the computed surface pressures on a NACA0012 aerofoil at α = 0, 5, and 10 ; note that the axis on the enclosed figure is greatly expanded in the region of the leading edge and pressure minimum. C p, the pressure coefficient, is defined as the gauge pressure at the position x, P P 0, where the latter is the free-stream static pressure) divided by the free-stream dynamic pressure: Aerofoils, Lift, Drag and Circulation 4

5 Figure 4.7: Computed pressure distribution around a NACA 0012 aerofoil. C p P P = (4.5) 1 ρ U and we have followed aerodynamic convention by plotting negative (lift-producing) C p upwards. Unfortunately the pressure coefficient has the same symbol as the turbine s power coefficient, but the former is used only in this Section. Note that there are two values of C P for each x/c; the (usually) negative value for the upper surface and the other for the lower surface. It is easy to show that the area between the upper and lower surface values gives the lift; note the coincidence of the values for α = 0. These simple calculations do not include the effects of viscosity, and so apply, in principle, only at infinite Re, but are sufficiently representative to indicate the acceleration of the flow over the upper surface that becomes increasingly rapid as α increases. The boundary layers on the upper and lower surfaces begin at the stagnation point where C p = 1. As Re decreases, these boundary layers remain laminar for a longer distance along the aerofoil surface. After the minimum pressure point on the upper surface, the boundary layer faces a region of adverse pressure gradient, which it can withstand more readily when turbulent (at higher Re) but not in its low-re laminar state. The usual result is separation and the formation of a laminar separation bubble; soon after the flow separates, it goes through transition to turbulence, and reattaches, often quickly. However, the bubble reduces the lift and increases drag. It can be readily seen from Exercise 1.9, that very small (micro) turbines have micro-reynolds numbers of around 10,000 or less at low wind speed. There is very little data available on aerofoil performance at these conditions partly because it is difficult to measure accurately the small forces involved. Laitone (1997) and others suggest that the best aerofoils at Re < 100,000 are very thin less than about 2.5% - and have around 5% camber. The thinness can be a major problem, because it becomes detrimental as the Re increases as the blades speed up to their rated output where the Re can be nearly 200,000, see Exercise At these Re, optimum performance requires around 10-15% thickness. 5 Aerofoils, Lift, Drag and Circulation

6 Figure 4.8: Lift and drag coefficients for high incidence. 4.3 Aerofoil Characteristics at High Angle of Attack The other situation for which there is a distinct lack of data is at high angles of attack. It is generally held that these data are needed primarily for vertical axis wind turbines, but it is often the case that the tip twist angle of a well-designed horizontal-axis blade is close to zero degrees. If the blade is stationary then its tip angle of attack is close to 90 so that performance at high α is important for the analysis of starting behaviour. Ostawari & Naik (1984) measured the lift and drag of a number of the NACA four digit aerofoils, and Figure 4.8 was constructed by digitising their published data. It is emphasised that the Figure is intended only as a qualitative guide to the overall behaviour of aerofoils like the NACA One interesting result is that c l remains high after stall at around α 15, at least until α reaches 60 and more. At α = 90, any reasonably thin aerofoil should behave more or less as a two-dimensional, thin flat plat held normally to the flow for which c l 0 and c d 2. We will look more fully at lift and drag at high incidence in Chapter 6 where starting performance is analysed. 4.4 The Effects of Thickness and Camber Table 4.1 shows that the SG series of aerofoils provides a range of thickness and camber, so it is useful to compare the lift and drag characteristics of other members of the family with those shown in Figures , remembering that the SG6040 is, primarily, a root aerofoil. Increasing the camber increases c l at α = 0, without greatly altering the slope of the linear part of the c l curve. What is particularly noticeable about the drag results for the SG6040 (greatest thickness) - see Figure and SG6043 (greatest camber) see Figure is the local maxima appearing in c d in the otherwise low-drag region at Re = 100,000. This is usually the result of a laminar separation bubble, which will usually be forced to burst early by further increases in α. Aerofoils, Lift, Drag and Circulation 6

7 Figure 4.9: Lift coefficient for SG6041 aerofoil. Figure 4.10: Drag coefficient for SG6041 aerofoil. Figure 4.11: Drag polars for SG6041 aerofoil. Figure 4.12: Drag polars for SG6041 aerofoil. 7 Aerofoils, Lift, Drag and Circulation

8 Figure 4.13: Lift coefficient for SG6043 aerofoil. Figure 4.14: Drag coefficient for SG6043 aerofoil. Figure 4.15: Drag polars for SG6043 aerofoil Figure 4.16: Drag polars for SG6043 aerofoil. Aerofoils, Lift, Drag and Circulation 8

9 4.5 The Circulation We return to the idea introduced in Section 3.1 of considering the two-dimensional analogue of the axisymmetric blade elements at radius r. This analogue is sketched in Figure 4.17, where only four of the infinite cascade of aerofoils is shown. In terms of the blade element chord, c, the non-dimensional spacing between the elements is just the inverse of the solidity, σ. Now we assume for simplicity that the aerofoil produces only lift, which must have a component down the page to produce blade rotation and we now consider the circulation, Γ, taken in an anti-clockwise circuit around the dotted rectangle which we divide into four legs. Furthermore, the analysis is made easier if the radius r is also the radius of the streamtube intersecting that element. As a result, legs 1 and 3 must be close to the blade. Figure 4.17: Circulation around a cascade of blade elements at radius r. The circulation is defined for any closed contour by Γ = U. dl where dl is the increment along the curve, and has the units of velocity length or m 2 /sec in the SI system. Thus the circulation around a circular contour centred on the turbine s axis (in a plane parallel to that of the blades) is 2πrW where W is the circumferential velocity. This shows the close connection between Γ and angular momentum. One of the other very useful properties of Γ, is that by Gauss theorem it is equal to the area integral of the vorticity within the contour, which is normally confined to the boundary layers on the upper and lower surfaces of the blade. The first leg in Figure 4.17 is upstream of the blades. If it is also a component of the circular contour mentioned above, then the first leg cannot contribute to Γ because the wind is assumed steady, one-dimensional and spatially uniform, and hence inviscid, and the circulation can never be changed in an inviscid fluid. (This simple and elegant argument was first proposed in 1915 by the famous British aerodynamicist G.I. Taylor, Taylor (1915).) Now the second leg is traversed from upstream to downstream whereas the fourth leg is traversed in the opposite direction. If we are careful to make the width of the contour equal to σ 1, then the contribution to Γ from legs 2 and 4 will cancel. Thus Γ is determined entirely from the third leg and we can write (4.6) 9 Aerofoils, Lift, Drag and Circulation

10 NΓ = 2πrW (4.7) where W is the average circumferential velocity in the wake. If we combine this equation with (3.5) and (3.10) for zero drag, then we obtain l = ρu T Γ (4.8) If the blades are stationary, then U T U 0 as σ 0, and we arrive at l = ρu 0 Γ (4.9) which is the famous Kutta-Joukowski equation for aerofoils. Circulation around aerofoils manifests itself as the so-called upwash upstream and the downwash downstream of the aerofoil. It is interesting that the derivation of (4.9) for aerofoils is actually much more difficult than this proof for a cascade, see, for example, Batchelor (1967, Section 6.4). However, the derivation also highlights one major difference between aerofoils and the circumferentially-constrained blade elements: whereas no circumferential velocity can be induced upstream of blades, there is no such constraint on aerofoils or indeed, on a cascade of aerofoils. In other words, there can be no net upwash for wind turbine blades. The assumption that half the far-wake circumferential velocity defines the value of a for the velocity triangle in Figure 3.2, must, therefore, be viewed as an attempt to redistribute W so as to produce an upwash for blade elements. Aerofoil behaviour can, therefore, only be an approximation for blade elements at sufficiently small values of W in the wake; typically W increases as σ increases. Apart from its connection with angular momentum, two useful features of Γ are that it measures vortex strength and it is a conserved quantity in an otherwise inviscid flow; if the blades have a bound vorticity of strength Γ, then the strength of the vortices trailing from the blades is also Γ. Note that it is the circulation that is conserved, not the vorticity. The significance of conservation of circulation is that (as we will see in Chapter 5) the simplest wake structure, giving a uniform U and leading to the Lanchester-Betz limit, is when the N hub vortices lie along the turbine axis and the N tip vortices are constant diameter helices in the far-wake. These trailing vortices are analogous to those on an aircraft wing, and arise from the same cause: the conservation of circulation. In other words, the bound vorticity of the blades or wings cannot end at the tips, and so must be converted into trailing vorticity. The effects of helical trailing vorticity on the flow over the blades is, however, much more difficult to calculate than that of the nearly straight wing tip vortices. One way to achieve this wake structure, and, therefore to have a turbine whose performance can approach the Lanchester-Betz limit, is to have the bound circulation of the blades nearly constant. For reasonably high values of λ, U T Ω r, and combining Equation (4.8) with the basic definition of lift coefficient, c l, leads to Γ ½Ω r c l c (4.10) so that if c l remains roughly constant, c must decrease with radius, as shown in Figure 3.4. This decrease is a feature of all well-designed wind turbine blades. Furthermore, the higher the value of λ, the smaller c needs to be. It is also a feature of efficient blades that solidity decreases with increasing operational λ. In Chapters 5 and 6 we will look at some typical magnitudes of Γ, and in the latter Chapter, see how its value is fixed for optimum turbine performance. Aerofoils, Lift, Drag and Circulation 10

11 4.6 Further Reading A good description of the early NACA work on aerofoil design is Abbott & von Doehoff (1959). General information on the fundamentals of aerofoils with aeronautical application can be found in books such as Bertin & Smith (1998) and McCormick (1995). Information on lift and drag of aerofoils can be found in Althaus (1996), Althaus & Wortmann (1981), Miley (1982), and from the web site maintained by Professor Michael Selig s group at the University of Illinois, Urbana-Champaign: Hard copies of their data can be found in Selig et al. (1989, 1996), and Lyon et al. (1997). References ABBOTT, I. H. and VON DOENHOFF, A.E. (1959). Theory of Wing Sections, Dover. ALTHAUS, D. (1996). Niedriggeschwindigkeitsprofile : Profilentwicklungen und Polarenmessugen im Laminarwindkanal des Instututs fã¼r Aerodynamik und Gasdynamik der Universität Stuttgart, Braunschweig, F. Vieweg. ALTHAUS, D. and WORTMANN, F. X. (1981). Stuttgarter Profilkatalog, F. Vieweg. BATCHELOR, G. K. (1967). An Introduction to Fluid Dynamics, Cambridge University Press. BERTIN, J. J. and SMITH, M. L. (1998). Aerodynamics for Engineers, Prentice-Hall International, 3 rd eds. GIGUERE, P. and SELIG, M. S. (1997). Low Reynolds Number Airfoils for Small Horizontal-Axis Wind Turbines, Wind Engineering, 21, GIGUERE, P. and SELIG, M. S. (1998). New Airfoils for Small Horizontal-Axis Wind Turbines, Transactions A.S.M.E., Journal of Solar Energy Engineering, 120, LAITONE, E. V. (1997). Aerodynamic lift at Reynolds numbers below , American Inst. Aeronautics & Astronautics Journal, 34, LYON, C. A.; BROEREN, A. P.; GIGUERE, P.; GOPALARATHNAM, A. and SELIG, M. S. (1997). Summary of Low-Speed Airfoil Data, 3, Soartech Publ.. McCORMICK, B. W. (1995). Aerodynamics, Aeronautics, and Flight Mechanics, John Wiley & Sons, 2 nd eds. McCROSKEY, W. J. (1987). A Critical Assessment of Wind Tunnel Results for the NACA0012 Airfoil, NASA Technical Memorandum MILEY, S. J. (1982). A Catalog of Low Reynolds Number Airfoil Data for Wind Turbine Applications, Report DE , U.S. Dept Energy. OSTOWARI, C. and NAIK, D. (1984). Post Stall Characteristics of Untwisted Varying Aspect Ratio Blades with an NACA 4415 Airfoil Section, Wind Engineering, 8, SELIG, M. S.; DONOVAN, J. F. and FRASER, D.B. (1989). Airfoils at Low Speeds, Soartech Publ. SELIG, M. S.; LYON, C. A.; GIGUERE, P.; NINHAM, C.P. and GUGLIELMO, J.J. (1996). Summary of Low-speed Airfoil Data, 2, Soartech Publ. TAYLOR, G. I. The Rotational Inflow Factor in Propeller Theory, Aeronautical Research Council (Great Britain) Reports & Memoranda 765, 1921 [reprinted in vol. 3 of G.K. Batchelor (ed.), The Scientific Papers of G.I. Taylor, Cambridge University Press, 1963.] 11 Aerofoils, Lift, Drag and Circulation

12 Exercises 1. The Vestas V80 turbine has a rated power output of 2 MW with a rotor diameter of 80 m and a maximum blade speed of 19 r.p.m. Assuming that the V80 has geometrically similar blades to those on V44 described in Chapter 1, use Equations (4.3) and (4.4) to estimate the improvement in efficiency caused by the increase in Re. 2. The three blades of a vertical axis wind turbine are attached to the hub by horizontal support arms, one for each blade. The turbine is 4 m in diameter and the chord length of the profiled but nonlifting supports is 7 cm. To determine the effect of the supports drag on performance, the blades were removed and the main shaft was instrumented to measure the torque, T, necessary to drive the supports at an angular velocity Ω (in rad/sec) in still air. The data were fitted by the curve T (Nm) = Ω 2. Show that this is consistent with the drag coefficient c d being independent of radius and determine its value. 3. Show that the pressure coefficient, C P, has the value of unity at the stagnation point of an aerofoil in incompressible flow. 4. Why can C P never exceed this value? Aerofoils, Lift, Drag and Circulation 12