Michael Graham - Wind Energy Handbook

Figure A3.16 Variation of the drag coefficient with Reynolds number at low angles of attack.

Characteristics for the NACA0012 aerofoil are shown in Figures A3.16and A3.17.

Cambered aerofoils, such as the NACA4412 shown in Figure A3.18, like cambered plates have curved mean lines, and this allows them to produce lift at zero angle of attack.

Generally, cambered aerofoils have their minimum drag range (drag bucket) at angles of attack well above zero. Thus, they are able to attain higher maximum lift/drag ratios than symmetrical aerofoils for positive angles of attack and useful lift coefficients, and this is the reason for their use.

The classification of the NACA four‐digit range of aerofoils, which were commonly used on earlier wind turbines, is very simple and is illustrated in Figure A3.19: from left to right, the first digit represents the amount of camber as a percentage of the chord length, the second digit represents the percentage chord position, in units of 10%, at which the maximum camber occurs, and the last two digits are the maximum thickness to chord ratio, as a percentage of the chord length, which, in this family of aerofoils, is at the 30% chord position. The cambered mean line, called the camber line , comprises two parabolic arcs that join smoothly at the point of maximum camber. For details of the extensive range of five‐ and six‐digit NACA aerofoils the reader should refer to Theory of Wing Sections by Abbott and von Doenhoff (1959).

Figure A3.17 Variation of the drag and lift coefficients with Reynolds number in the stall region.

Figure A3.18 The profile of the NACA4412 aerofoil.

The angle of attack α is measured from the chord line, which is now defined as the straight line joining the ends of the camber line.

Note that the lift at zero angle of attack is no longer zero; zero lift occurs at a small negative angle of attack. With most cambered aerofoils, the zero lift angle in degrees is approximately equal to ‐A°, where A is the percentage camber.

The behaviour of the NACA4412 aerofoil is shown in Figure A3.20for angles of attack below and just above the stall. Positive lift occurs at zero angle of attack. Zero lift occurs at a small negative angle of attack of approximately −4°.

The centre of pressure (i.e. the point at which the lift acts), which is at the quarter‐chord position on symmetrical aerofoils, lies aft of the quarter‐chord position on cambered aerofoils and moves towards the leading edge with increasing angle of attack until the stall. After the stall the centre of pressure on all aerofoils moves rearward towards the midchord. However, if forces are evaluated with reference to a fixed chordwise position, then the resultant force through this point is accompanied by a pitching moment about this point (nose‐up positive, by convention). The reference point is usually the quarter‐chord point (c/4 back from the leading edge), but sometimes it is the midchord and sometimes the torsion axis of the section. If a pitching moment coefficient is defined as

(A3.8)

Figure A3.19 Classification of the NACAXXXX aerofoil range.

Figure A3.20 The characteristics of the NACA4412 aerofoil for Re = 1.5·10 6.

then there will be a position, called the aerodynamic centre , for which . Theoretically, the aerodynamic centre lies at the quarter‐chord position and is close to this point for most practical aerofoils.

The value of C mdepends upon the degree of camber, but for the NACA4412 the value is −0.1. Note that pitching moments are always negative in practice (nose down) despite the sign convention.

Above the stall the pre‐stall position of the aerodynamic centre usually continues to be used, although no longer satisfying the above definition.

1 1There is not complete agreement on the appropriate name for this limit because Lanchester (1915), Betz (1919, 1920), and Joukowski (1920) all published relevant papers. Betz limit is the most commonly used name. See also comment in Chapter 1.

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