Inviscid, Incompressible Flow Past Thin Airfoils

3.1 Introduction

The solution of the inviscid, incompressible and irrotational (potential) two-dimensional flow past an arbitrary profile with a sharp trailing edge exists and can be obtained using complex variables and mapping techniques. However, this approach is very difficult in general, except for particular classes of profiles, such as the Joukowski airfoils.

A more practical approach consists in simplifying the model one step further by assuming what is called a “thin airfoil”, an airfoil that creates only a small disturbance to the uniform incoming flow. It will then be possible to simplify the tangency con­dition, linearize the expression of the pressure coefficient and construct the solution of the flow past an arbitrary airfoil, using superposition of elementary solutions.

3.1.1 Definition of a Thin Airfoil

The thin airfoil geometry is defined with the profile chord aligned with the x-axis, from x = 0 to x = c, which corresponds to zero incidence. The (smooth) leading

© Springer Science+Business Media Dordrecht 2015 51

J. J. Chattot and M. M. Hafez, Theoretical and Applied Aerodynamics,

DOI 10.1007/978-94-017-9825-9_3

edge is tangent to the z-axis at the origin, as in Fig. 3.2. The sharp trailing edge is located at (c, 0).

The equations of the upper and lower surfaces of the airfoil are given by f + (x) and f -(x), respectively. The camber distribution d(x) and thickness distribution e(x) > 0 are defined as follows

d(x) = 2 (f + (x) + f -(x) , e(x) = f + (x) – f -(x) (3.1)

Note that, with the above definition of the airfoil geometry, f ±(0) = f ±(c) = 0. Hence, d(0) = e(0) = d(c) = e(c) = 0. With these distributions, the profile upper and lower surfaces are obtained from the superposition of camber and thickness as

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f +(x) = d(x) + 2e(x), f (x) = d(x) – 2e(x) (3.2)

The curve z = d (x) is called the mean camber line. Let d be its maximum absolute value. The maximum value of e(x), e is called maximum thickness.

The relative camber of the airfoil is d and | is the relative thickness.

A thin airfoil is defined as an airfoil with small relative camber and relative thickness

de

– << 1, – << 1 (3.3)

cc

The Wright Brothers glider wings can be considered as equipped with thin airfoils as can be seen in Fig. 3.1.