Page 176 - Aerodynamics for Engineering Students

Two-dimensional wing theory

4.1 Introduction

By the end of the nineteenth century the theory of ideal, or potential, flow (see
Chapter 3) was extremely well-developed. The motion of an inviscid fluid was a well-
defined mathematical problem. It satisfied a relatively simple linear partial differen-
tial equation, the Laplace equation (see Section 3.2), with well-defined boundary
conditions. Owing to this state of affairs many distinguished mathematicians were
able to develop a wide variety of analytical methods for predicting such flows. Their
work was and is very useful for many practical problems, for example the flow around
airships, ship hydrodynamics and water waves. But for the most important practical
applications in aerodynamics potential flow theory was almost a complete failure.
Potential flow theory predicted the flow field absolutely exactly for an inviscid
fluid, that is for infinite Reynolds number. In two important respects, however, it did
not correspond to the flow field of real fluid, no matter how large the Reynolds
number. Firstly, real flows have a tendency to separate from the surface of the body.
This is especially pronounced when the bodies are bluff like a circular cylinder, and in
such cases the real flow bears no resemblance to the corresponding potential flow.
Secondly, steady potential flow around a body can produce no force irrespective of
the shape. This result is usually known as d’Alembert’s paradox after the French
mathematician who first discovered it in 1744. Thus there is no prospect of using