Page 263 - Aerodynamics for Engineering Students

246 Aerodynamics for Engineering Students

Thus
1 1
CL-PV'S
L = A~T-~V~~S~
=
2 2
and writing aspect ratio (AR) = 481s gives
CL TA~ (AR) (5.47)
This indicates the rather surprising result that the lift depends on the magnitude of
the coefficient of the first term only, no matter how many more may be present in the
series describing the distribution. This is because the terms A3 sin 38, As sin 58, etc.,
provide positive lift on some sections and negative lift on others so that the overall
effect of these is zero. These terms provide the characteristic variations in the
spanwise distribution but do not affect the total lift of the whole which is determined
solely from the amplitude of the first harmonic. Thus

CL = T(AR)AI and L = 27rpV2?A1 (5.47a)

Down wash
Changing the variable and limits of Eqn (5.32), the equation for the downwash is

w0, =-
47rs s" case - COS el
In this case I? = 4sV A, sin n8 and thus on differentiating
dr
dB=4sVxnA,cosn8

Introducing this into the integral expression gives

= nA,G,
7r
and writing in G, = nsinn8l/sin81 from Appendix 3, and reverting back to the
general point 8:
nA, sin ne
w=v (5.48)
sin 8
This involves all the coefficients of the series, and will be symmetrically distributed
about the centre line for odd harmonics.

Induced drag (vortex drag)
The drag grading is given by d, = pwr. Integrating gives the total induced drag

D, = Lpwrdz
or in the polar variable