Page 58 - Aerodynamics for Engineering Students

Basic concepts and definitions 41

;

L!.///
p"
I I

+-
V

Fig. 1.19
The total pitching moment coefficient is

CM = CMZ + CMX (1.65)
In Fig. 1.20 are shown the graphs necessary for the evaluation of the aerodynamic
coefficients for the mid-section of a three-dimensional wing with an ellipto-
Zhukovsky profile.

1.5.7 Induced drag

Section 5.5 below should also be referred to. Consider what is happening at some
point y along the wing span (Fig. 1.21). Each of the trailing vortices produces a
downwards component of velocity, w, at y, known as the downwash or induced
velocity (see Section 5.5.1). This causes the flow over that section of the wing to
be inclined slightly downwards from the direction of the undisturbed stream V
(Fig. 1.22) by the angle E, the induced angle of incidence or downwash angle. The
local flow is also at a slightly different speed, q.
If the angle between the aerofoil chord line and the direction of the undisturbed
stream, the geometric angle of incidence, is a, it is seen that the angle between the
chord line and the actual flow at that section of the wing is equal to a-E, and this is
called the effective incidence am. It is this effective incidence that determines the lift
coefficient at that section of the wing, and thus the wing is lifting less strongly than
the geometric incidence would suggest. Since the circulation and therefore w and E
increase with lift coefficient, it follows that the lift of a three-dimensional wing
increases less rapidly with incidence than does that for a two-dimensional wing,
which has no trailing vortices.
Now the circulation round this section of the wing will have a value r appro-
priate to a,, and the lift force corresponding to this circulation will be pqr per
unit length, acting perpendicular to the direction of q as shown, i.e. inclined
backwards from the vertical by the angle E. This force therefore has a component
perpendicular to the undisturbed stream V, that, by definition, is called the lift,
and is of magnitude
V
I = pqr cos E = pqr - = pVr per unit length
4
There is also a rearwards component of magnitude