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Potential flow 105

studying potential flows. Were we interested only in bluff bodies like circular cylin-
ders there would indeed be little point in studying potential flow, since no matter how
high the Reynolds number, the real flow around a circular cylinder never looks
anything like the potential flow. (But that is not to say that there is no point in
studying potential flow around a circular cylinder. In fact, the study of potential flow
around a rotating cylinder led to the profound Kutta-Zhukovski theorem that links
lift to circulation for all cross-sectional shapes.) But potential flow really comes into
its own for slender or streamlined bodies at low angles of incidence. In such cases the
boundary layer remains attached until it reaches the trailing edge or extreme rear of
the body. Under these circumstances a wide low-pressure wake does not form, unlike
a circular cylinder. Thus the flow more or less follows the shape of the body and the
main viscous effect is the generation of skin-friction drag plus a much smaller
component of form drag.
Potential flow is certainly useful for predicting the flow around fuselages and other
non-lifting bodies. But what about the much more interesting case of lifting bodies
like wings? Fortunately, almost all practical wings are slender bodies. Even so there is
a major snag. The generation of lift implies the existence of circulation. And circul-
ation is created by viscous effects. Happily, potential flow was rescued by an important
insight known as the Kuttu condition. It was realized that the most important effect of
viscosity for lifting bodies is to make the flow leave smoothly from the trailing edge.
This can be ensured within the confines of potential flow by conceptually placing one
or more (potential) vortices within the contour of the wing or aerofoil and adjusting
the strength so as to generate just enough circulation to satisfy the Kutta condition.
The theory of lift, i.e. the modification of potential flow so that it becomes a suitable
model for predicting lift-generating flows is described in Chapters 4 and 5.

3.1.1 The velocity potential

The stream function (see Section 2.5) at a point has been defined as the quantity
of fluid moving across some convenient imaginary line in the flow pattern, and lines of
constant stream function (amount of flow or flux) may be plotted to give a picture
of the flow pattern (see Section 2.5). Another mathematical definition, giving a
different pattern of curves, can be obtained for the same flow system. In this case
an expression giving the amount of flow along the convenient imaginary line is found.
In a general two-dimensional fluid flow, consider any (imaginary) line OP joining
the origin of a pair of axes to the point P(x, y). Again, the axes and this line do not
impede the flow, and are used only to form a reference datum. At a point Q on the
line let the local velocity q meet the line OP in /3 (Fig. 3.1). Then the component of
velocity parallel to 6s is q cos p. The amount of fluid flowing along 6s is q cos ,6 6s. The
total amount of fluid flowing along the line towards P is the sum of all such amounts
and is given mathematically as the integral Jqcospds. This function is called the
velocity potential of P with respect to 0 and is denoted by 4.
Now OQP can be any line between 0 and P and a necessary condition for
Sqcospds to be the velocity potential 4 is that the value of 4 is unique for the
point P, irrespective of the path of integration. Then:

Velocity potential q5 = q cos /3 ds (3.1)
LP
If this were not the case, and integrating the tangential flow component from 0 to P
via A (Fig. 3.2) did not produce the same magnitude of 4 as integrating from 0 to P