Page 192 - Aerodynamics for Engineering Students

Two-dimensional wing theory 175

and subtracting

-
p2-p1=-pu 2[ 2 ("1 --- "2) + (32-(?)2]
2 uu
and with the aerofoil thin and at small incidence the perturbation velocity ratios ul/U
are
and u2/U will be so small compared with unity that (u1/U)2 and (u~/U)~ neglected
compared with ul/U and uZ/U, respectively. Then
P = p2 - P1 = PWUl - u2) (4.19)
The equivalent vorticity distribution indicates that the circulation due to element
Ss is kSx (Sx because the camber line deviates only slightly from the Ox axis).
Evaluating the circulation around 6,s and taking clockwise as positive in this case,
by taking the algebraic sum of the flow of fluid along the top and bottom of Ss, gives

kSx = +(U + u~)SX - (U + UZ)SX = (~1 - u~)SX (4.20)
Comparing (4.19) and (4.20) shows that p = pUk as introduced in Eqn (4.17).
For a trailing edge angle of zero the Kutta condition (see Section 4.1.1) requires
u1 = 2.42 at the trailing edge. It follows from Eqn (4.20) that the Kutta condition is
satisfied if
k=O at x=c (4.21)

The induced velocity v in Eqn (4.14) can be expressed in terms of k, by considering
the effect of the elementary circulation k Sx at x, a distance x - x1 from the point
considered (Fig. 4.13). Circulation kSx induces a velocity at the point XI equal to

1 k6x
27rX-X1

from Eqn (4.5).
The effect of all such elements of circulation along the chord is the induced velocity
v' where

Fig. 4.13 Velocities at x1 from 0: U + u1, resultant tangential to camber lines; v', induced by chordwise
variation in circulation; U, free stream velocity inclined at angle Q to Ox