Induced Angle of Attack

Notice in Fig. 6.10 that the lifting line is now exposed to a relative wind, V, which is the vector sum of VM and the downwash, w. Thus, the local airfoil section behaves as if the geometric angle of attack, a, has changed by an amount a;, the induced angle of attack. This is the viewpoint that is carried forward—namely, that the angle of attack of the wing section was modified due to the presence of the vortex sheet.

Referring to Fig. 6.10, the concept of relative wind at the lifting line in Fig. 6.9 may be generalized to include the wing-section angle of attack and camber.

The angle of attack to which the wing section responds is no longer the geo­metric angle of attack, a, but rather the (smaller) effective angle of attack, aeff. Camber may be included in the section shape by introducing the zero-lift angle, aL0. Two-dimensional airfoil results may be used to specify aL0 for a particular wing section. When the line ZLL (see Fig. 6.10) is aligned with VM the lift of the airfoil section (behaving two-dimensionally) is zero. If a finite wing is set to a zero-lift condition, then a wing zero lift line (ZLLW) may be specified that is identical all across the span. A wing angle of attack, aaw, then can be defined as the angle between the freestream and the ZLLW.

Now, we recall from two-dimensional thin-airfoil theory Eq. 5.74 that the lift coefficient of a thin airfoil is given by:

Cl = mo(a — aLo), (6.11)

where the two-dimensional lift-curve slope, m0 per radian, may be assigned the theoretical value of 2n or a value from experiment. Equation 6.11 also may be written with the lift-curve slope as a0, where m0 is per radian and a0 is per degree.

Because a section of a three-dimensional wing is assumed to behave locally as a two-dimensional airfoil, it follows that for a finite-wing section:*

Cl = mo(aeff – aLo). (6.12)

In some texts, the quantity (aeff — aLo) is written as Oq.

Finally, from the geometry shown in Fig. 6.12, at every spanwise station:

aeff = a – Oj, (6.13)

where aj is a positive angle. From Fig. 6.11:

w

tan aj = – —= a-. (6.14)

V ж [23]

Recall that the downwash, w, is a downward-directed velocity component in Car­tesian coordinates and is negative. Hence, a minus sign must be introduced in Eq. 6.14 so that the induced angle of attack, aj, is a positive angle. In general, the induced angle of attack varies across the span.

The mathematical expression for the induced angle of attack follows from Eq. 6.14 and the equation for downwash, Eq. 6.10: