Page 193 - Aerodynamics for Engineering Students

176 Aerodynamics for Engineering Students

and introducing this in Eqn (4.14) gives

(4.22)

The solution for kdx that satisfies Eqn (4.22) for a given shape of camber
line (defining dy,/dx) and incidence can be introduced in Eqns (4.17) and (4.18) to
obtain the lift and moment for the aerofoil shape. The characteristics CL and Cv,,
follow directly and hence kCp, the centre of pressure coefficient, and the angle
for zero lift.

*(
4.4 The solution of the generat equation

In the general case Eqn (4.22) must be solved directly to determine the function k(x)
that corresponds to a specified camber-line shape. Alternatively, the inverse design
problem may be solved whereby the pressure distribution or, equivalently, the
tangential velocity variation along the upper and lower surfaces of the aerofoil is
given. The corresponding k(x) may then be simply found from Eqns (4.19) and
(4.20). The problem then becomes one of finding the requisite camber line shape
from Eqn (4.22). The present approach is to work up to the general case through the
simple case of the flat plate at incidence, and then to consider some practical
applications of the general case. To this end the integral in Eqn (4.22) will be
considered and expressions for some useful definite integrals given.
In order to use certain trigonometric relationships it is convenient to change
variables from x to 8, through x = (c/2)(1 - cos Q), and to HI, then the limits
change as follows:

Q~OAT as xwO+c, and

so

kdx j7 ksinOd0 (4.23)
1 -k (cos8 - cosQ1)

Also the Kutta condition (4.21) becomes
k=O at Q=T (4.24)

The expressions found by evaluating two useful definite integrals are given below
cosnQ sin nO1
sin 81
s (COS Q - COS 6'1) dQ = ny : n = 0,1,2,. . . (4.25)
0
sinnQsinQ
dQ=-rcosnQI :n=0,1,2, ...
s (COS Q - COS 01) (4.26)
0
The derivations of these results are given in Appendix 3. However, it is not necessary
to be familiar with this derivation in order to use Eqns (4.25) and (4.26) in applica-
tions of the thin-aerofoil theory.