Page 232 - Aerodynamics for Engineering Students

Finite wing theory 21 5

to the two-dimensional flow plane considered previously and the influence of the
so-called line vortex is the influence, in a section plane, of an infinitely long, straight-line
vortex of vanishingly small area.
In general, the vortex axis will be a curve in space and area Swill have finite size. It
is convenient to assume that S is made up of several elemental areas or, alternatively,
that the vortex consists of a bundle of elemental vortex lines or filaments. Such
a bundle is often called a vortex tube (c.f. a stream tube which is a bundle of
streamlines), being a tube bounded by vortex filaments.
Since the vortex axis is a curve winding about within the fluid, capable of flexure
and motion as a whole, the estimation of its influence on the fluid at large is some-
what complex and beyond the present intentions. All the vortices of significance to
the present theory are fixed relative to some axes in the system or free to move in
a very controlled fashion and can be assumed to be linear. Nonetheless, the vortices
will not all be of infinite length and therefore some three-dimensional or end influ-
ence must be accounted for.
Vortices conform to certain laws of motion. A rigorous treatment of these is
precluded from a text of this standard but may be acquired with additional study
of the basic references.*

5.2.1 Helmholtz’s theorems
The four fundamental theorems of vortex motion in an inviscid flow are named after
their author, Helmholtz. The first theorem has been discussed in part in Sections 2.7
and 4.1, and refers to a fluid particle in general motion possessing all or some of the
following: linear velocity, vorticity, and distortion. The second theorem demon-
strates the constancy of strength of a vortex along its length. This is sometimes
referred to as the equation of vortex continuity. It is not difficult to prove that the
strength of a vortex cannot grow or diminish along its axis or length. The strength of
a vortex is the magnitude of the circulation around it and this is equal to the product
of the vorticity C and area S. Thus
r = ~s

It follows from the second theorem that CS is constant along the vortex tube (or
filament), so that if the section area diminishes, the vorticity increases and vice versa.
Since infinite vorticity is unacceptable the cross-sectional area S cannot diminish to
zero.
In other words a vortex line cannot end in the fluid. In practice the vortex line must
form a closed loop, or originate (or terminate) in a discontinuity in the fluid such as
a solid body or a surface of separation. A refinement of this is that a vortex tube
cannot change in strength between two sections unless vortex filaments of equivalent
strength join or leave the vortex tube (Fig. 5.6). This is of great importance in the
vortex theory of lift.
The third and fourth theorems demonstrate respectively that a vortex tube consists
of the same particles of fluid, i.e. there is no fluid interchange between tube and
surrounding fluid, and the strength of a vortex remains constant as the vortex moves
through the fluid.
The theorem of most consequence to the present chapter is theorem two, although
the third and fourth are tacitly accepted as the development proceeds.

* Saffman, P.G. 1992 Vortex Dynamics, Cambridge University Press.