Thickness problem for thin-aerofoil theory

Before extending the theory to take account of the thickness of aerofoil sections, it is useful to review the parts of the method. Briefly, in thin-aerofoil theory, above, the two-dimensional thin wing is replaced by the vortex sheet which occupies the camber surface or, to the first approximation, the chordal plane. Vortex filaments comprising the sheet extend to infinity in both directions normal to the plane, and all velocities are confined to the xy plane. In such a situation, as shown in Fig. 4.12, the sheet supports a pressure difference producing a normal (upward) increment of force of (p — pi)8s per unit spanwise length. Suffices 1 and 2 refer to under and upper sides of the sheet respectively. But from Bernoulli’s equation:

P ~ Pi = ^ p{u ~ u]) = p{u2 – щ) Ul 2 U’ (4.93)

Writing («2 + wi)/2 — U the free-stream velocity, and г/2 – u = к, the local loading on the wing becomes

ІР — pi)Ss — pUk6s (4.94)

The lift may then be obtained by integrating the normal component and similarly the pitching moment. It remains now to relate the local vorticity to the thin shape of the aerofoil and this is done by introducing the solid boundary condition of zero velocity normal to the surface. For the vortex sheet to simulate the aerofoil completely, the velocity component induced locally by the distributed vorticity must be sufficient to make the resultant velocity be tangential to the surface. In other words, the compon­ent of the free-stream velocity that is normal to the surface at a point on the aerofoil must be completely nullified by the normal-velocity component induced by the distributed vorticity. This condition, which is satisfied completely by replacing the surface line by a streamline, results in an integral equation that relates the strength of the vortex distribution to the shape of the aerofoil.

So far in this review no assumptions or approximations have been made, but thin – aerofoil theory utilizes, in addition to the thin assumption of zero thickness and small camber, the following assumptions:

(a) That the magnitude of total velocity at any point on the aerofoil is that of the local chordwise velocity = U + u1.

(b) That chordwise perturbation velocities г/ are small in relation to the chordwise component of the free stream U.

(c) That the vertical perturbation velocity v anywhere on the aerofoil may be taken as that (locally) at the chord.

Making use of these restrictions gives

– fCk dx

Jo 2tt x – ,n


and thus Eqn (4.42) is obtained:

• ‘4Ус _ 1 _ Г

.d* J Jo

k cbc

27Г X — X

 

(Eqn (4.42))

 

U

 

This last integral equation relates the chordwise loading, i. e. the vorticity, to the shape and incidence of the thin aerofoil and by the insertion of a suitable series expression for k in the integral is capable of solution for both the direct and indirect aerofoil problems. The aerofoil is reduced to what is in essence a thin lifting sheet, infinitely long in span, and is replaced by a distribution of singularities that satisfies the same conditions at the boundaries of the aerofoil system, i. e. at the surface and at infinity. Further, the theory is a linearized theory that permits, for example, the velocity at a point in the vicinity of the aerofoil to be taken to be the sum of the velocity components due to the various characteristics of the system, each treated separately. As shown in Section 4.3, these linearization assumptions permit an extension to the theory by allowing a perturbation velocity contribution due to thickness to be added to the other effects.

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