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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 a₀, called the lift‐curve slope , is about 6.0 (∼0.1/deg.).

Note that a₀ should 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 α₀ is the angle of attack for zero lift and proportional to the camber, being negative for positive camber (convex upwards).

Therefore

with a₀ = 2π.

Generally, thickness increases a₀ and 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_l with the angle of attack α is shown in Figure A3.13 for 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).