## Finite wing theory

Preamble

Whatever the operating requirements of an aeroplane may be in terms of speed; endurance, pay-load and so on, a critical stage in its eventual operation is in the low-speed flight regime, and this must be accommodated in the overall design process. The fact that low-speed flight was the classic flight regime has meant that over the years a vast array of empirical data has been accumulated from flight and other tests, and a range of theories and hypotheses set up to explain and extend these observations. Some theories have survived to provide successful working processes for wing design that are capable of further exploitation by computational methods.

In this chapter such a classic theory is developed to the stage of initiating the preliminary low-speed aerodynamic design of straight, swept and delta wings. Theoretical fluid mechanics of vortex systems are employed, to model the loading properties of lifting wings in terms of their geometric and attitudinal characteristics and of the behaviour of the associated flow processes.

The basis on which historical solutions to the finite wing problem were arrived at are explained in detail and the work refined and extended to take advantage of more modern computing techniques.

A great step forward in aeronautics came with the vortex theory of a lifting aerofoil due to Lanchester* and the subsequent development of this work by Prandtl.t Previously, all aerofoil data had to be obtained from experimental work and fitted to other aspect ratios, planforms, etc., by empirical formulae based on past experience with other aerofoils.

Among other uses the Lanchester-Prandtl theory showed how knowledge of two-dimensional aerofoil data could be used to predict the aerodynamic characteristics of (three-dimensional) wings. It is this derivation of the aerodynamic characteristics of wings that is the concern of this chapter. The aerofoil data can either be obtained empirically from wind-tunnel tests or by means of the theory described in Chapter 4. Provided the aspect ratio is fairly large and the assumptions of thin-aerofoil theory are met (see Section 4.3 above), the theory can be applied to wing planforms and sections of any shape.

* see Bibliography.

' Prandtl, L. (1918), Tragflugellheorie, Nachr. Ges. Wiss., Gotlingen, 107 and 451.

### 5.1 The vortex system

Lanchester's contribution was essentially the replacement of the lifting wing by a theoretical model consisting of a system of vortices that imparted to the surrounding air a motion similar to the actual flow, and that sustained a force equivalent to the lift known to be created. The vortex system can be divided into three main parts: the starting vortex; the trailing vortex system; and the bound vortex system. Each of these may be treated separately but it should be remembered that they are all component parts of one whole.

### 5.1.1 The starting vortex

When a wing is accelerated from rest the circulation round it, and therefore the lift, is not produced instantaneously. Instead, at the instant of starting the streamlines over the rear part of the wing section are as shown in Fig. 5.1, with a stagnation point occurring on the rear upper surface. At the sharp trailing edge the air is required to change direction suddenly while still moving at high speed. This high speed calls for extremely high local accelerations producing very large viscous forces and the air is unable to turn round the trailing edge to the stagnation point. Instead it leaves the surface and produces a vortex just above the trailing edge. The stagnation point moves towards the trailing edge, the circulation round the wing, and therefore its lift, increasing progressively as the stagnation point moves back. When the stagnation point reaches the trailing edge the air is no longer required to flow round the trailing edge. Instead it decelerates gradually along the aerofoil surface, comes to rest at the trailing edge, and then accelerates from rest in a different direction (Fig. 5.2). The vortex is left behind at the point reached by the wing when the stagnation point

reached the trailing edge. Its reaction, the circulation round the wing, has become stabilized at the value necessary to place the stagnation point at the trailing edge (see Section 4.1.1).* The vortex that has been left behind is equal in strength and opposite in sense to the circulation round the wing and is called the starting vortex or initial eddy.

### 5.1.2 The trailing vortex system

The pressure on the upper surface of a lifting wing is lower than that of the surrounding atmosphere, while the pressure on the lower surface is greater than that on the upper surface, and may be greater than that of the surrounding atmosphere. Thus, over the upper surface, air will tend to flow inwards towards the root from the tips, being replaced by air that was originally outboard of the tips. Similarly, on the undersurface air will either tend to flow inwards to a lesser extent, or may tend to flow outwards. Where these two streams combine at the trailing edge, the difference in spanwise velocity will cause the air to roll up into a number of small streamwise vortices, distributed along the whole span. These small vortices roll up into two large vortices just inboard of the wing-tips. This is illustrated in Fig. 5.3. The strength of

### Fig. 5.3 The horseshoe vortex

* There is no fully convincing physical explanation for the production of the starting vortex and the generation of the circulation around the aerofoil. Various incomplete explanations will be found in the references quoted in the bibliography. The most usual explanation is based on the large viscous forces associated with the high velocities round the trailing edge, from which it is inferred that circulation cannot be generated, and aerodynamic lift produced, in an inviscid fluid. It may be, however, that local flow acceleration is equally important and that this is sufficiently high to account for the failure of the flow to follow round the sharp trailing edge, without invoking viscosity. Certainly it is now known, from the work of T. Weis-Fogh [Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production, J. Expl. Biol., 59, 169-230, 1973] and M.J. Lighthill [On the Weis-Fogh mechanism of lift generation, J. Fluid Mech., 60,1-17,1973] on the hovering flight of the small wasp Ertcarsia formosa, that it is possible to generate circulation and lift in the complete absence of viscosity.

In practical aeronautics, fluid is not inviscid and the complete explanation of this phenomenon must take account of viscosity and the consequent growth of the boundary layer as well as high local velocities as the motion is generated.

Fig. 5.3 The horseshoe vortex each of these two vortices will equal the strength of the vortex replacing the wing itself.

The existence of the trailing and starting vortices may easily be verified visually. When a fast aeroplane pulls out of a dive in humid air the reduction of pressure and temperature at the centres of the trailing vortices is often sufficient to cause some of the water vapour to condense into droplets, which are seen as a thin streamer for a short distance behind each wing-tip (see frontispiece).

To see the starting vortex all that is needed is a tub of water and a small piece of board, or even a hand. If the board is placed upright into the water cutting the surface and then suddenly moved through the water at a moderate incidence, an eddy will be seen to leave the rear, and move forwards and away from the 'wing'. This is the starting vortex, and its movement is induced by the circulation round the plate.

### 5.1.3 The bound vortex system

Both the starting vortex and the trailing system of vortices are physical entities that can be explored and seen if conditions are right. The bound vortex system, on the other hand, is a hypothetical arrangement of vortices that replace the real physical wing in every way except that of thickness, in the theoretical treatments given in this chapter. This is the essence of finite wing theory. It is largely concerned with developing the equivalent bound vortex system that simulates accurately, at least a little distance away, all the properties, effects, disturbances, force systems, etc., due to the real wing.

Consider a wing in steady flight. What effect has it on the surrounding air, and how will changes in basic wing parameters such as span, planform, aerodynamic or geometric twist, etc., alter these disturbances? The replacement bound vortex system must create the same disturbances, and this mathematical model must be sufficiently flexible to allow for the effects of the changed parameters. A real wing produces a trailing vortex system. The hypothetical bound vortex must do the same. A consequence of the tendency to equalize the pressures acting on the top and bottom surfaces of an aerofoil is for the lift force per unit span to fall off towards the tips. The bound vortex system must produce the same grading of lift along the span.

For complete equivalence, the bound vortex system should consist of a large number of spanwise vortex elements of differing spanwise lengths all turned backwards at each end to form a pair of the vortex elements in the trailing system. The varying spanwise lengths accommodate the grading of the lift towards the wing-tips, the ends turned back produce the trailing system and the two physical attributes of a real wing are thus simulated.

For partial equivalence the wing can be considered to be replaced by a single bound vortex of strength equal to the mid-span circulation. This, bent back at each end, forms the trailing vortex pair. This concept is adequate for providing good estimations of wing effects at distances greater than about two chord lengths from the centre of pressure.

### 5.1.4 The horseshoe vortex

The total vortex system associated with a wing, plus its replacement bound vortex system, forms a complete vortex ring that satisfies all physical laws (see Section 5.2.1). The starting vortex, however, is soon left behind and the trailing pair stretches effectively to infinity as steady flight proceeds. For practical purposes the system consists of the bound vortices and the trailing vortex on either side close to the wing. This three-sided vortex has been called the horseshoe vortex (Fig. 5.3).

Study of the completely equivalent vortex system is largely confined to investigating wing effects in close proximity to the wing. For estimation of distant phenomena the system is simplified to a single bound vortex and trailing pair, known as the simplified horseshoe vortex (Fig. 5.4). This is dealt with in Section 5.3, before the more involved and complete theoretical treatments of wing aerodynamics.

### 5.2 Laws of vortex motion

The theoretical modelling of the flow around wings was discussed in the previous section. There the use of an equivalent vortex system to model the lifting effects of a wing was described. In order to use this theoretical model to obtain quantitative predictions of the aerodynamic characteristics of a wing it is necessary first to study the laws of vortex motion. These laws also act as a guide for understanding how modern computationally based wing theories may be developed.

In the analysis of the point vortex (Chapter 3) it was considered to be a string of rotating particles surrounded by fluid at large moving irrotationally under the influence of the rotating particles. Further, the flow investigation was confined to a plane section normal to the length or axis of the vortex. A more general definition is that a vortex is a flow system in which a finite area in a normal section plane contains vorticity. Figure 5.5 shows the section area S of a vortex so called because S possesses vorticity. The axis of the vortex (or of the vorticity, or spin) is clearly always normal r

Fig. 5.5 The vorticity of a section of vortex tube to the two-dimensional flow plane considered previously and the influence of the so-called line vortex is the influence, in a section plane, of an infinitely long, straight-line vortex of vanishingly small area.

In general, the vortex axis will be a curve in space and area S will have finite size. It is convenient to assume that S is made up of several elemental areas or, alternatively, that the vortex consists of a bundle of elemental vortex lines or filaments. Such a bundle is often called a vortex tube (c.f. a stream tube which is a bundle of streamlines), being a tube bounded by vortex filaments.

Since the vortex axis is a curve winding about within the fluid, capable of flexure and motion as a whole, the estimation of its influence on the fluid at large is somewhat complex and beyond the present intentions. All the vortices of significance to the present theory are fixed relative to some axes in the system or free to move in a very controlled fashion and can be assumed to be linear. Nonetheless, the vortices will not all be of infinite length and therefore some three-dimensional or end influence must be accounted for.

Vortices conform to certain laws of motion. A rigorous treatment of these is precluded from a text of this standard but may be acquired with additional study of the basic references.*

5.2.1 Helmholtz's theorems

The four fundamental theorems of vortex motion in an inviscid flow are named after their author, Helmholtz. The first theorem has been discussed in part in Sections 2.7 and 4.1, and refers to a fluid particle in general motion possessing all or some of the following: linear velocity, vorticity, and distortion. The second theorem demonstrates the constancy of strength of a vortex along its length. This is sometimes referred to as the equation of vortex continuity. It is not difficult to prove that the strength of a vortex cannot grow or diminish along its axis or length. The strength of a vortex is the magnitude of the circulation around it and this is equal to the product of the vorticity £ and area S. Thus r = cs

It follows from the second theorem that (S is constant along the vortex tube (or filament), so that if the section area diminishes, the vorticity increases and vice versa. Since infinite vorticity is unacceptable the cross-sectional area S cannot diminish to zero.

In other words a vortex line cannot end in the fluid. In practice the vortex line must form a closed loop, or originate (or terminate) in a discontinuity in the fluid such as a solid body or a surface of separation. A refinement of this is that a vortex tube cannot change in strength between two sections unless vortex filaments of equivalent strength join or leave the vortex tube (Fig. 5.6). This is of great importance in the vortex theory of lift.

The third and fourth theorems demonstrate respectively that a vortex tube consists of the same particles of fluid, i.e. there is no fluid interchange between tube and surrounding fluid, and the strength of a vortex remains constant as the vortex moves through the fluid.

The theorem of most consequence to the present chapter is theorem two, although the third and fourth are tacitly accepted as the development proceeds.

* Saffman, P.G. 1992 Vortex Dynamics, Cambridge University Press.

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5.2.2 The Biot-Savart law

The original application of this law was in electromagnetism, where it relates the intensity of the magnetic field in the vicinity of a conductor carrying an electric current to the magnitude of the current. In the present application velocity and vortex strength (circulation) are analogous to the magnetic field strength and electric current respectively, and a vortex filament replaces the electrical conductor. Thus the Biot-Savart law can also be interpreted as the relationship between the velocity induced by a vortex tube and the strength (circulation) of the vortex tube. Only the fluid motion aspects will be further pursued here, except to remark that the term induced velocity, used to describe the velocity generated at a distance by the vortex tube, was borrowed from electromagnetism.

Allow a vortex tube of strength T, consisting of an infinite number of vortex filaments, to terminate in some point P. The total strength of the vortex filaments will be spread over the surface of a spherical boundary of radius R (Fig. 5.7) as the filaments diverge from the point P in all directions. The vorticity in the spherical surface will thus have the total strength T.

Owing to symmetry the velocity of flow in the surface of the sphere will be tangential to the circular line of intersection of the sphere with a plane normal to the axis of the vortex. Moreover, the direction will be in the sense of the circulation about the vortex. Figure 5.8 shows such a circle ABC of radius r subtending a conical angle of 26 at P. If the velocity on the sphere at R, 6 from P is v, then the circulation round the circuit ABC is T' where

The net velocity in the circuit ABC is the sum of Eqns (5.4) and (5.5):

<YKT

A-kt

As Pi approaches P

47Tr

47Tr

This is the induced velocity at a point in the field of an elementary length 6s of vortex of strength T that subtends an angle 66 at P located by the coordinates R, 6 from the element. Since r = .Rsinfl and R66 = it is more usefully quoted as:

### Special cases of the Biot-Savart law

Equation (5.6) needs further treatment before it yields working equations. This treatment, of integration, varies with the length and shape of the finite vortex being studied. The vortices of immediate interest are all assumed to be straight lines, so no shape complexity arises. They will vary only in their overall length.

A linear vortex of finite length AB Figure 5.10 shows a length AB of vortex with an adjacent point P located by the angular displacements a and ¡3 from A and B respectively. Point P has, further, coordinates r and 6 with respect to any elemental length 6s of the length AB that may be defined as a distance s from the foot of the perpendicular h. From Eqn (5.7) the velocity at P induced by the elemental length 6s is

6v = -—^sinfl&y 4irrl in the sense shown, i.e. normal to the plane APB.

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