Page 235 - Aerodynamics for Engineering Students

21 8 Aerodynamics for Engineering Students
The net velocity in the circuit ABC is the sum of Eqns (5.4) and (5.5):
r
v-vl =-~i-cose-(~-cose~)~
47rr
n

As PI approaches P
COS el -+ cOs(8 - be) = COS e + sin e 66
and
v - VI --f sv

giving
r
Sv = -sine68
47rr
This is the induced velocity at a point in the field of an elementary length 6s of vortex
of strength r that subtends an angle 68 at P located by the coordinates R, 8 from the
element. Since r = R sin 0 and R 60 = 6s sin 0 it is more usefully quoted as:

sin
sv = - ess (5-7)
47rR2

Special cases of the Biot-Savart law
Equation (5.6) needs further treatment before it yields working equations. This
treatment, of integration, varies with the length and shape of the finite vortex being
studied. The vortices of immediate interest are all assumed to be straight lines, so no
shape complexity arises. They will vary only in their overall length.
A linear vortex of fuzite length AB Figure 5.10 shows a length AB of vortex with an
adjacent point P located by the angular displacements o and p from A and B
respectively. Point P has, further, coordinates r and 0 with respect to any elemental
length 6s of the length AB that may be defined as a distance s from the foot of the
perpendicular h. From Eqn (5.7) the velocity at P induced by the elemental length 6s is
r
6v = --sin86s (5.8)
4nr2
in the sense shown, i.e. normal to the plane APB.

Fig. 5.10