Thin Aerofoil Theory

6.1 Introduction

The main limitation of Joukowski’s theory is that it is applicable only to the Joukowski family of aerofoil sections. Similar is the case with aerofoils obtained with other transformations. These aerofoils do not permit a satisfactory solution of the reverse problem of aerofoil design, that is, to start with the loading distribution and from the loading, obtain the necessary aerofoil profile. For the indirect or reverse solution to be possible, a theory which consists of more local relationships is required. That is:

• The overall lifting property of a two-dimensional aerofoil depends on the circulation it generates and this, for the far-field or overall effects, has been assumed to be concentrated at a point within the aerofoil profile, and to have a magnitude related to the incidence, camber and thickness of the aerofoil.

• The loading on the aerofoil, or the chordwise pressure distribution, follows as a consequence of the parameters, namely the incidence, camber and thickness. But the camber and thickness imply a characteristic shape which depends in turn on the conformal transformation function and the basic flow to which it is applied.

• The profiles obtained with Joukowski transformation do not lend themselves to modern aerofoil design.

• However, Joukowski transformation is of direct use in aerofoil design. It introduces some features which are the basis to any aerofoil theory, such as:

(a) The lift generated by an aerofoil is proportional to the circulation around the aerofoil profile, that is, L а Г.

(b) The magnitude of the circulation Г must be such that it keeps the velocity finite in the vicinity of trailing edge.

• It is not necessary to concentrate the circulation in a single vortex, as shown in Figure 6.1(a), and an immediate extension to the theory is to distribute the vorticity throughout the region surrounded by the aerofoil profile in such a way that the sum of the distributed vorticity equals that of the original model, as shown in Figure 6.1(b), and the vorticity at the trailing edge is zero.

This mathematical model may be simplified by distributing the vortices on the camber line and disre­garding the effect of thickness. In this form it becomes the basis for the classical “thin aerofoil theory” of Munk and Glauert.

2

Considering the fact that the transformation Z = г + Z applied to a circle in an uniform stream gives a straight line aerofoil (that is, a flat plate), the theory assumed that the general thin aerofoil could be

Theoretical Aerodynamics, First Edition. Ethirajan Rathakrishnan.

© 2013 John Wiley & Sons Singapore Pte. Ltd. Published 2013 by John Wiley & Sons Singapore Pte. Ltd.

a camber line.

replaced by its camber line, which is assumed to be only a slight distortion of a straight line. Consequently the shape from which the camber line has to be transformed would be a similar distortion from the original circle. The original circle could be found by transforming the slightly distorted shape, shown in Figure 6.2.

This transformation function defines the distortion, or change of shape, of the circle, and hence by implication, the distortion (or camber) of the straight line aerofoil. As shown in Figure 6.2, the circle (z = ae’e) inthez-plane is transformed to the “S” shape in the z’-plane using the transformation z! = f (z), and then the S shape to a cambered profile using the following transformation:

Z = z’ + –

z

и

= f (z) + 7Й’

It is evident that z’ = f (z) defines the shape of the camber and Glauert used the series expansion:

z’ = z 1 +

for this. Using potential theory and Joukowski hypothesis, the lift and pitching moment acting on the aerofoil section were found in terms of the coefficients Ax, that is in terms of the shape parameter.

The usefulness or advantage of the theory lies in the fact that the aerofoil characteristics could be quoted in terms of the coefficient Ax , which in turn could be found by graphical integration method from any camber line.