Finite Aerofoil Theory

8.1 Introduction

The vortex theory of a lifting aerofoil proposed by Lancaster and the subsequent development by Prandtl made use of for calculating the forces and moment about finite aerofoils. The vortex system around a finite aerofoil consists of the starting vortex, the trailing vortex system and the bound vortex system, as illustrated in Figure 8.1.

The horseshoe vortex system around an aerofoil, consisting of the bound and trailing vortices, can be simplified as shown in Figure 8.2.

8.2 Relationship between Spanwise Loading and Trailing Vorticity

From Helmholtz’s second theorem, the strength of the circulation around any section of a bundle of vortex tubes is the sum of the strength of the vortex filaments cut by the section plane. As per this theorem, the spanwise variation of the strength of the combined bound vortex filaments may be shown as illustrated in Figure 8.3.

If the circulation curve can be described as some function of y, say f (y), then the strength of the circulation shed by the aerofoil becomes:

that is:

Sk = – f'(y) dy. (8.1)

Now at a section of the aerofoil the lift per unit span is given by:

l = pUk,

where p and U are the density and velocity of the freestream. Thus, for a given flight speed and flow density, the circulation strength k is proportional to l. From the above discussion, it can be inferred that:

• The trailing filaments are closer showing that the vorticity strength is larger near the wing tips than other locations. That is, near the wing tips, the vorticity content of the vortices shed are very strong.

• Aerofoils with infinite span (b ^ to) or two-dimensional aerofoils will have constant spanwise loading.

Theoretical Aerodynamics, First Edition. Ethirajan Rathakrishnan.

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