Michael Graham - Wind Energy Handbook

The lift coefficient is defined as

(A3.3)

U is the flow speed and A is the plan area of the body. For a long body, such as an aircraft wing or a wind turbine blade, the lift per unit span is used in the definition, the plan area now being taken as the chord length (multiplied by unit span):

(A3.4)

Figure A3.12 Stalled flow around an aerofoil.

In practice it is convenient to write for pre‐stall conditions:

(A3.5)

where a 0, called the lift‐curve slope , is about 6.0 (∼ 0.1/deg .).

Note that a 0should not be confused with the flow induction factor.

Thin aerofoil potential flow theory shows that for a flat plate or very thin aerofoil, the Kutta–Joukowski condition is satisfied by

where α 0is the angle of attack for zero lift and proportional to the camber, being negative for positive camber (convex upwards).

Therefore

with a 0= 2π.

Generally, thickness increases a 0and viscous effects (the boundary layer) decrease it.

Lift, therefore, depends on two parameters, the angle of attack α and the flow speed U . The same lift force can be generated by different combinations of α and U .

The variation of C lwith the angle of attack α is shown in Figure A3.13for a typical symmetrical aerofoil (NACA0012). Notice that the simple relationship of Eq. (A3.5)is only valid for the pre‐stall region, where the flow is attached. Because the angle of attack is small (< 16° ) the equation is often simplified to

(A3.6)

Figure A3.13 C l − α curve for a symmetrical aerofoil.

The potential velocity and pressure field around aerofoil sections may be calculated using transformation theory (classically) or more usually now by the boundary integral panel method. Many commercial CFD codes also include an option to compute the potential flow calculation by field methods (finite difference, volume or element). The resulting potential flow solution may then be made more realistic, taking into account the effects of the laminar and/or turbulent boundary layers by using the potential flow results for the surface pressure or velocity to drive boundary layer calculations of displacement thickness that in turn modify the potential flow as well as providing estimates of drag. Direct methods are used for this while the flow remains unseparated but inverse methods must be used as separation develops. These methods are very efficient computationally and give good results up to angles of attack at which a shallow separation has started the aerofoil stall. Once a large separation has developed (full stall), they become less accurate, and CFD methods (discussed in Chapter 4) must be used. The well‐known code XFOIL (Drela 1989) is a widely used example of this type of method. These techniques are discussed in more detail in Katz and Plotkin (1991).

The definition of the drag coefficient for a streamlined body, such as an aircraft wing or a wind turbine blade, because of the relevance of surface friction drag is based not on the frontal area but on the plan area. The flow past a body that has a large span normal to the flow direction is locally quasi‐2‐D, and in such cases the drag coefficient can be based upon the drag force per unit span using the streamwise chord length for the definition:

(A3.7)

The drag coefficient of an aerofoil varies with angle of attack. For a well‐designed aerofoil at moderate to high Reynolds number [O(10 6)–O(10 7)], the value of C dis O( 0.01 ) in the minimum drag range of angle of attack (called the drag bucket ).

The following sections show some results for two classical NACA four‐digit aerofoils that, although not now used except exceptionally for wind turbines, do demonstrate the typical force behaviour of aerofoil sections.

Figure A3.11shows that on the upper surface pressure is rising as the flow moves from the suction peak towards the trailing edge. This is an adverse pressure gradient that slows the air down. It also thickens the boundary layer more rapidly, causing more velocity momentum to be lost. If the flow above (i.e. just off) the surface within the boundary layer is slowed to a standstill, the surface streamlines separate from the surface, stall occurs, and the pressure drag rises sharply. The strength of the adverse pressure gradient increases with angle of attack, and therefore the drag will also rise with angle of attack. Figure A3.14shows the variation of C dwith α for the symmetrical NACA0012 aerofoil.

Figure A3.14 Variationof C dwith α for the NACA0012 aerofoil.

Figure A3.15 Lift/drag ratio variation for the NACA0012 aerofoil.

The efficiency of a wind turbine rotor blade is significantly affected by the lift/drag ratio of its aerofoil section(s) (as shown in Figure A3.15), and it is desirable that a turbine blade operates at the maximum ratio point.

The nature of the flow pattern around an aerofoil is determined by the Reynolds number, and this affects the values of the lift and drag coefficients. The general level of the drag coefficient increases with decreasing Reynolds number. The effect on the lift coefficient is largely concerned with the angle of attack at which stall occurs. Below a critical Reynolds number of about 200 000, the boundary layer remains laminar, usually leading to early stall or partial separation (long bubbles) at low angles of attack.. As the Reynolds number rises, so does the stall angle and, because the lift coefficient increases linearly with angle of attack below the stall, the maximum value of the lift coefficient also rises.