Page 194 - Aerodynamics for Engineering Students

Two-dimensional wing theory 177

4.4.1 The thin symmetrical flat plate aerofoil
In this simple case the camber line is straight along Ox, and dy,/dx = 0. Using
Eqn (4.23) the general equation (4.22) becomes

(4.27)

What value should k take on the right-hand side of Eqn (4.27) to give a left-hand side
which does not vary with x or, equivalently, e? To answer this question consider the
result (4.25) with n = 1. From this it can be seen that

Comparing this result with Eqn (4.27) it can be seen that if k = kl = 2Ua cos f3/sin f3
it will satisfy Eqn (4.27). The only problem is that far from satisfying the Kutta
condition (4.24) this solution goes to infinity at the trailing edge. To overcome this
problem it is necessary to recognize that if there exists a function k2 such that

(4.28)

then k = kl + k2 will also satisfy Eqn (4.27).
Consider Eqn (4.25) with n = 0 so that
1
sT (cose-cosel) de = 0
Comparing this result to Eqn (4.28) shows that the solution is

where C is an arbitrary constant.
Thus the complete (or general) solution for the flat plate is given by
2uacose+ c
k = kl +kz =
sin 8
The Kutta condition (4.24) will be satisfied if C = 2Ua giving a final solution of

(4.29)

Aerodynamic coefficients for a flat plate
The expression for k can now be put in the appropriate equations for lift and moment
by using the pressure:
1 +case
p = pUk = 2pU2a (4.30)
sin 0