# Two-dimensional wing theory

Preamble

Here the basic fluid mechanics outlined previously is applied to the analysis of the flow about a lifting wing section. It is explained that potential flow theories of themselves offer little further scope for this problem unless modified to simulate certain effects of real flows. The result is a powerful but elementary aerofoil theory capable of wide exploitation. This is derived in the general form and applied to a number of discrete aeronautical situations, including the flapped aerofoil and the jet flap. The ‘reverse’ problem is also presented: to determine the rudimentary aerofoil shape that produces certain aerodynamic performance requirements. This theory is essentially relevant to thin aerofoils but thickness parameters are added to enhance the practical applications of the method. Classical mathematical solutions are referred to, also the solutions offered towards the end of the chapter that employ computational panel methods.

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 differen­tial 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

potential flow theory in its pure form to estimate the lift or drag of wings and thereby to develop aerodynamic design methods.

Flow separation and d’Alembert’s paradox both result from the subtle effects of viscosity on flows at high Reynolds number. The necessary understanding and knowledge of viscous effects came largely from work done during the first two decades of the twentieth century. It took several more decades, however, before this knowledge was fully exploited in aerodynamic design. The great German aeronaut­ical engineer Prandtl and his research team at the University of Gottingen deserve most of the credit both for explaining these paradoxes and showing how potential flow theory can be modified to yield useful predictions of the flow around wings and thus of their aerodynamic characteristics. His boundary-layer theory explained why flow separation occurs and showed how skin-friction drag could be calculated. This theory and its later developments are described in Chapter 7 below. He also showed how a theoretical model based on vortices could be developed for the flow field of a wing having large aspect ratio. This theory is described in Chapter 5. There it is shown how a knowledge of the aerodynamic characteristics, principally the lift coefficient, of a wing of infinite span – an aerofoil – can be adapted to give estimates of the aerodynamic characteristics of a wing of finite span. This work firmly estab­lished the relevance of studying the two-dimensional flow around aerofoils that is the subject of the present chapter.