Page 105 - Aerodynamics for Engineering Students

88 Aerodynamics for Engineering Students

' Bundle of vortex tubes
Fig. 2.27

Circulation can be regarded as a measure of the combined strength of the total
number of vortex lines passing through A. It is a measure of the vorticityjlux carried
through A by these vortex lines. The relationship between circulation and vorticity is
broadly similar to that between momentum and velocity or that between internal
energy and temperature. Thus circulation is the property of the region A bounded by
control surface C, whereas vorticity is a flow variable, like velocity, defined at a
point. Strictly it makes no more sense to speak of conservation, generation, or
transport of vorticity than its does to speak of conservation, generation, or transport
of velocity. Logically these terms should be applied to circulation just as they are to
momentum rather than velocity. But human affairs frequently defy logic and aero-
dynamics is no exception. We have become used to speaking in terms of conservation
etc. of vorticity. It would be considered pedantic to insist on circulation in this
context, even though this would be strictly correct. Our only motivation for
making such fine distinctions here is to elucidate the meaning and significance of
circulation. Henceforth we will adhere to the common usage of the terms vorticity
and circulation.
In two-dimensional flow, in the absence of the effects of viscosity, circulation is
conserved. This can be expressed mathematically as follows:

(2.82)

In view of what was written in Section 2.7.6 about the link between vorticity and
viscous effects, it may seem somewhat illogical to neglect such effects in Eqn (2.82).
Nevertheless, it is often a useful approximation to use Eqn (2.82).
Circulation can also be evaluated by means of an integration around the
perimeter C. This can be shown elegantly by applying Stokes theorem to Eqn
(2.8 1); thus

(2.83)

This commonly serves as the definition of circulation in most aerodynamics text.
The concept of circulation is central to the theory of lift. This will become clear in
Chapters 5 and 6.