Page 57 - Aerodynamics for Engineering Students

40 Aerodynamics for Engineering Students
or, simply

CZ = f C,cosEd(s/c) = f Cpd(x/c), (1.59b)

where the contour integral is evaluated by following an anti-clockwise direction
around the contour C of the aerofoil.
Similar arguments lead to the following relations for X.
6Xu = pubs sin E, 6Xe = pe6s sin E: 6s sin E = 6z,

giving

where zmu and zme are respectively the maximum and minimum values of z, and AC,
is the difference between the values of C, acting on the fore and rear points of an
aerofoil for a fixed value of z.
The pitching moment can also be calculated from the pressure distribution. For
simplicity, the pitching moment about the leading edge will be calculated. The
contribution due to the force 62 acting on a slice of aerofoil of length 6x is given by
6~4 (Pu -pe)xbx = [(Pu - pm) - (Po - pm)lx6x;
=
so, remembering that the coefficient of pitching moment is defined as
M M
-
CM=- - in this case, as S = c,
ipv2sc $pv2c2
the coefficient of pitching moment due to the Z force is given by

(1.61)

Similarly, the much smaller contribution due to the X force may be obtained as

(1.62)

The integrations given above are usually performed using a computer or graphically.
The force coefficients CX and CZ are parallel and perpendicular to the chord line,
whereas the more usual coefficients CL and CD are defined with reference to the
direction of the free-stream air flow. The conversion from one pair of coefficients to
the other may be carried out with reference to Fig. 1.19, in which, CR, the coefficient
of the resultant aerodynamic force, acts at an angle y to CZ. CR is both the resultant
of CX and CZ, and of CL and CD; therefore from Fig. 1.19 it follows that
CL = CR COS(^ + a) = CR COS y COS Q - CR sin y sin a
But CR cosy = CZ and CR sin y = Cx, so that
CL = CZ cosa - Cxsina. (1.63)
Similarly