Page 249 - Aerodynamics for Engineering Students

232 Aerodynamics for Engineering Students

vortex filaments cut by the section plane. As the section plane is progressively moved
outwards from the centre section to the tips, fewer and fewer bound vortex filaments
are left for successive sections to cut so that the circulation around the sections
diminishes. In this way, the spanwise change in circulation round the wing is related
to the spanwise lengths of the bound vortices. Now, as the section plane is moved
outwards along the bound bundle of filaments, and as the strength of the bundle
decreases, the strength of the vortex filaments so far shed must increase, as the overall
strength of the system cannot diminish. Thus the change in circulation from section
to section is equal to the strength of the vorticity shed between these sections.
Figure 5.21 shows a simple rectangular wing shedding a vortex trail with each pair
of trailing vortex filaments completed by a spanwise bound vortex. It will be noticed
that a line joining the ends of all the spanwise vortices forms a curve that, assuming
each vortex is of equal strength and given a suitable scale, would be a curve of the
total strengths of the bound vortices at any section plotted against the span. This
curve has been plotted for clarity on a spanwise line through the centre of pressure of
the wing and is a plot of (chordwise) circulation (I') measured on a vertical ordinate,
against spanwise distance from the centre-line (CL) measured on the horizontal
ordinate. Thus at a section z from the centre-line sufficient hypothetical bound
vortices are cut to produce a chordwise circulation around that section equal to I'.
At a further section z + Sz from the centre-line the circulation has fallen to l? - ST,
indicating that between sections z and z + Sz trailing vorticity to the strength of
SI' has been shed.
If the circulation curve can be described as some function of z,flz) say then the
strength of circulation shed

(5.25)

Now at any section the lift per span is given by the Kutta-Zhukovsky theorem
Eqn (4.10)
I=pVT
and for a given flight speed and air density, I' is thus proportional to 1. But I is the
local intensity of lift or lift grading, which is either known or is the required quantity
in the analysis.
The substitution of the wing by a system of bound vortices has not been rigorously
justified at this stage. The idea allows a relation to be built up between the physical
load distribution on the wing, which depends, as shall be shown, on the wing
geometric and aerodynamic parameters, and the trailing vortex system.
Figure 5.21 illustrates two further points:
(a) It will be noticed from the leading sketch that the trailing filaments are closer
together when they are shed from a rapidly diminishing or changing distribution
curve. Where the filaments are closer the strength of the vorticity is greater. Near
the tips, therefore, the shed vorticity is the most strong, and at the centre where
the distribution curve is flattened out the shed vorticity is weak to infinitesimal.
(b) A wing infinitely long in the spanwise direction, or in two-dimensional flow, will
have constant spanwise loading. The bundle will have filaments all of equal
length and none will be turned back to form trailing vortices. Thus there is no
trailing vorticity associated with two-dimensional wings. This is capable of
deduction by a more direct process, i.e. as the wing is infinitely long in the
spanwise direction the lower surface @ugh) and upper surface (low) pressures