Category Aerodynamics for Engineering Students

A start is made by considering the influence of the end effect, or downwash, on the lifting properties of an aerofoil section at some distance z from the centre-line of the wing. Figure 5.34 shows the lift-versus-incidence curve for an aerofoil section of

Fig. 5.34 Lift-versus-incidence curve for an aerofoil section of a certain profile, working two-dimensionally and working in a flow regime influenced by end effects, i. e. working at some point along the span of a finite lifting wing

a certain profile working two-dimensionally and working in a flow regime influenced by end effects, i. e. working at some point along the span of a finite lifting wing. Assuming that both curves are linear over the range considered, i. e. the working range, and that under both flow regimes the zero-lift incidence is the same, then

(5.56) (5.57)

(5.58)

and since

This is Prandtl’s integral equation for the circulation Г at any section along the span in terms of all the aerofoil parameters. These will be discussed when Eqn (5.59) is reduced to a form more amenable to numerical solution. To do this the general series expression (5.45) for Г is taken:

r = 45F]T An sin пв The previous section gives Eqn (5.48):

VEnA„ sin пв

V Y nAn sin пв

cax ‘ ‘ sin в

Cancelling V and collecting cax/Bs into the single parameter ц this equation becomes:

ЭО

ц{а – a0) = J An sinn6»(l +^j)

Я=1

The solution of this equation cannot in general be found analytically for all points along the span, but only numerically at selected spanwise stations and at each end.

This is the direct problem broadly facing designers who wish to predict the perform­ance of a projected wing before the long and costly process of model tests begin. This does not imply that such tests need not be carried out. On the contrary, they may be important steps in the design process towards a production aircraft.

The problem can be rephrased to suggest that the designers would wish to have some indication of how the wing characteristics vary as, for example, the geometric parameters of the project wing are changed. In this way, they can balance the aerodynamic effects of their changing ideas against the basic specification – provided there is a fairly simple process relating the changes in design parameters to the aerodynamic characteristics. Of course, this is stating one of the design problems in its baldest and simplest terms, but as in any design work, plausible theoretical processes yielding reliable predictions are very comforting.

The loading on the wing has already been described in the most general terms available and the overall characteristics are immediately to hand in terms of the coefficients of the loading distribution (Section 5.5). It remains to relate the coeffi­cients (or the series as a whole) to the basic aerofoil parameters of planform and aerofoil section characteristics.

Thus comparing Eqn (5.50) with the induced-drag coefficient for the elliptic case (Eqn (5.43)) it can be seen that modifying the spanwise distribution away from the elliptic increases the drag coefficient by the fraction 6 that is always positive. It follows that for the induced drag to be a minimum 6 must be zero so that the distribution for minimum induced drag is the semi-ellipse. It will also be noted that the minimum drag distribution produces a constant downwash along the span whereas all other distributions produce a spanwise variation in induced velocity. This is no coincidence. It is part of the physical explanation of why the elliptic distribution should have minimum induced drag.

To see this, consider two wings (Fig. 5.33a and b), of equal span with spanwise distributions in downwash velocity w = wo = constant along (a) and w = f(z) along

(b) . Without altering the latter downwash variation it can be expressed as the sum of two distributions wo and wi = fi(z) as shown in Fig. 5.33c. Fig. 5.33 (a) Elliptic distribution gives constant downwash and minimum drag, (b) Non-elliptic distribution gives varying downwash. (c) Equivalent variation for comparison purposes

The operations to obtain lift, downwash and drag vary only in detail from the previous cases.

Lift on the wing

and changing the variable z = — s cos в,

and substituting for the general series expression

The sum within the squared bracket equals zero for all values of n other than unity when it becomes

Thus

L = Anr^pV^s2 = Cj}-pV2S

and writing aspect ratio (AR) = As^jS gives

Cl = ttAi(AR) (5.47)

This indicates the rather surprising result that the lift depends on the magnitude of the coefficient of the first term only, no matter how many more may be present in the series describing the distribution. This is because the terms AzsinW, As sin 50, etc., provide positive lift on some sections and negative lift on others so that the overall effect of these is zero. These terms provide the characteristic variations in the spanwise distribution but do not affect the total lift of the whole which is determined solely from the amplitude of the first harmonic. Thus

CL = тг(АЛ)Аі and L = 2тгpV4Ax (5.47a)

Down wash

Changing the variable and limits of Eqn (5.32), the equation for the downwash is

cos в — COS 6

In this case Г = 4sV£A„ sinпв and thus on differentiating

dr

— = 4sV Y’ nA„ cos пв d в "

Introducing this into the integral expression gives

As V nA„ cos«0

COS в — COS Oi

= — y^nA„G„

7Г

and writing in G„ = 7rsinn0i/sin0i from Appendix 3, and reverting back to the general point в:

This involves all the coefficients of the series, and will be symmetrically distributed about the centre line for odd harmonics.

Induced drag (vortex drag)

The drag grading is given by dv = pwT. Integrating gives the total induced drag

pwTdz

or in the polar variable

This can be demonstrated by multiplying out the first three (say) odd harmonics, thus:

1= [ (A sin0 + ЗЛ3 sin30 + 5Л5 sin50)(.4i sin0 + A3 sin30 + As sin0)d0 Jo

= I {A sin2 0 + ЗЛ2 sin2 0 + 5A2 sin2 0 + [A 1A3 sin 0 sin 30 and Jo

other like terms which are products of different multiples of 0]} d0

On carrying out the integration from 0 to 7г all terms other than the squared terms vanish leaving

I = J (A2 sin2 0 + ЗЛ2 sin2 30 + SA sin2 50 H – )d0

= 2 f-4? + 3A + 5As ■!—- ] =

This gives

From Eqn (5.47)

cl

7Г2{ARf

and introducing this into Eqn (5.49)

Plainly 6 is always a positive quantity because it consists of squared terms that must always be positive. Cp, can be a minimum only when <5 = 0. That is when A3 = As = Aj = … = 0 and the only term remaining in the series is A sin 9.

In the previous section attention was directed to distributions of circulation (or lift) along the span in which the load is assumed to fall symmetrically about the centre-line according to a particular family of load distributions. For steady symmetric manoeuvres this is quite satisfactory and the previous distribution formula may be arranged to suit certain cases. Its use, however, is strictly limited and it is necessary to seek further for an expression that will satisfy every possible combination of wing design parameter and flight manoeuvre. For example, it has so far been assumed that the wing was an isolated lifting surface that in straight steady flight had a load distribution rising steadily from zero at the tips to a maximum at mid-span (Fig. 5.31a). The general wing, however, will have a fuselage located in the centre sections that will modify the loading in that region (Fig. 5.31b), and engine nacelles or other excrescences may deform the remainder of the curve locally.

The load distributions on both the isolated wing and the general aeroplane wing will be considerably changed in anti-symmetric flight. In rolling, for instance, the upgoing wing suffers a large decrease in lift, which may become negative at some incidences (Fig. 5.31c). With ailerons in operation the curve of spanwise loading for a wing is no longer smooth and symmetrical but can be rugged and distorted in shape (Fig. 5.3Id).

It is clearly necessary to find an expression that will accommodate all these various possibilities. From previous work the formula / = pVT for any section of span is familiar. Writing / in the form of the non-dimensional lift coefficient and equating to pVT:

(5.44)

is easily obtained. This shows that for a given steady flight state the circulation at any section can be represented by the product of the forward velocity and the local chord.

Now in addition the local chord can be expressed as a fraction of the semi-span s, and with this fraction absorbed in a new number and the numeral 4 introduced for later convenience, Г becomes:

Г = 4Crj

where Cr is dimensionless circulation which will vary similarly to Г across the span. In other words, Cr is the shape parameter or variation of the Г curve and being dimensionless it can be expressed as the Fourier sine series sin пв in which the coefficients A„ represent the amplitudes, and the sum of the successive harmonics describes the shape. The sine series was chosen to satisfy the end conditions of the curve reducing to zero at the tips where у = ±s. These correspond to the values of 9 = 0 and 7Г. It is well understood that such a series is unlimited in angular measure but the portions beyond 0 and 7Г can be disregarded here. Further, the series can fit any shape of curve but, in general, for rapidly changing distributions as shown by a rugged curve, for example, many harmonics are required to produce a sum that is a good representation.

In particular the series is simplified for the symmetrical loading case when the even terms disappear (Fig. 5.32 (II)). For the symmetrical case a maximum or minimum must appear at the mid-section. This is only possible for sines of odd values of 7t/2. That is, the symmetrical loading must be the sum of symmetrical harmonics. Odd

harmonics are symmetrical. Even harmonics, on the other hand, return to zero again at 7t/2 where in addition there is always a change in sign. For any asymmetry in the loading one or more even harmonics are necessary.

With the number and magnitude of harmonics effectively giving all possibilities the general spanwise loading can be expressed as

(5.45)

It should be noted that since / = pVT the spanwise lift distribution can be expressed as

(5.46)

The aerodynamic characteristics for symmetrical general loading are derived in the next subsection. The case of asymmetrical loading is not included. However, it may be dealt with in a very similar manner, and in this way expressions derived for such quantities as rolling and yawing moment.

In order to demonstrate the general method of obtaining the aerodynamic charac­teristics of a wing from its loading distribution the simplest load expression for symmetric flight is taken, that is a semi-ellipse. In addition, it will be found to be a good approximation to many (mathematically) more complicated distributions and is thus suitable for use as first predictions in performance estimates.

The spanwise variation in circulation is taken to be represented by a semi-ellipse having the span (2j) as major axis and the circulation at mid-span (Го) as the semi­minor axis (Fig. 5.30). From the general expression for an ellipse

or

(5.37)

This expression can now be substituted in Eqns (5.32), (5.34) and (5.36) to find the lift, downwash and vortex drag on the wing.

Fig. 5.30 Elliptic loading

Lift for elliptic distribution

From Eqn (5.34)

L = L, PVTil = //*T°V 1 “ ©2d*

i. e.

L = pVT 0тг-

whence

{pVirs)

or introducing

L = Cl-pV2S

giving the mid-span circulation in terms of the overall aerofoil lift coefficient and geometry.

Downwash for elliptic distribution

Here

Substituting this in Eqn (5.32)

w*, =^[tt + zi/] (5.40)

Now as this is a symmetric flight case, the shed vorticity is the same from each side of the wing and the value of the downwash at some point z is identical to that at the corresponding point – z on the other wing.

So substituting for ±zi in Eqn (5.40) and equating:

^=ё[*+г,,1=й[’г■z,,1

This identity is satisfied only if / = 0, so that for any point z — z along the span

Го

4s

This important result shows that the downwash is constant along the span.

Induced drag (vortex drag) for elliptic distribution

From Eqn (5.36)

^=/>“’rdz=/>Br»v/i-©5d2

— – = aspect ratio (Лі?)

Equation (5.43) establishes quantitatively how Cffy falls with a rise in (AR) and confirms the previous conjecture given above, Eqn (5.36), that at zero lift in sym­metric flight Сду is zero and the other condition that as (AR) increases (to infinity for two-dimensional flow) decreases (to zero).

The induced velocity at z is, in general, in a downwards direction and is sometimes called downwash. It has two very important consequences that modify the flow about the wing and alter its aerodynamic characteristics.

Firstly, the downwash that has been obtained for the particular point z is felt to a lesser extent ahead of z and to a greater extent behind (see Fig. 5.27), and has the effect of tilting the resultant oncoming flow at the wing (or anywhere else within its influence) through an angle where w is the local downwash. This reduces the effective incidence so that for the same lift as the equivalent infinite wing or two-dimensional wing at incidence ax an incidence a = аж + є is required at that section on the finite wing. This is illustrated in Fig. 5.28, which in addition shows how the two-dimensional lift Lis normal to

the resultant velocity VR and is, therefore, tilted back against the actual direction of motion of the wing V. The two-dimensional lift is resolved into the aerodynamic forces L and Dv respectively, normal to and against the direction of the forward velocity of the wing. Thus the second important consequence of downwash emerges. This is the generation of a drag force Dv. This is so important that the above sequence will be explained in an alternative way.

A section of a wing generates a circulation of strength Г. This circulation super­imposed on an apparent oncoming flow velocity V produces a lift force Loc = pVT according to the Kutta-Zhukovsky theorem (4.10), which is normal to the apparent oncoming flow direction. The apparent oncoming flow felt by the wing section is the resultant of the forward velocity and the downward induced velocity arising from the trailing vortices. Thus the aerodynamic force produced by the combination of Г and V appears as a lift force L normal to the forward motion and a drag force Dv against the normal motion. This drag force is called trailing vortex drag, abbreviated to vortex drag or more commonly induced drag (see Section 1.5.7).

Considering for a moment the wing as a whole moving through air at rest at infinity, two-dimensional wing theory suggests that, taking air as being of small to negligible viscosity, the static pressure of the free stream ahead is recovered behind the wing. This means roughly that the kinetic energy induced in the flow is converted back to pressure energy and zero drag results. The existence of a thin boundary layer and narrow wake is ignored but this does not really modify the argument.

In addition to this motion of the airstream, a finite wing spins the airflow near the tips into what eventually becomes two trailing vortices of considerable core size. The generation of these vortices requires a quantity of kinetic energy that is not recovered

by the wing system and that in fact is lost to the wing by being left behind. This constant expenditure of energy appears to the wing as the induced drag. In what follows, a third explanation of this important consequence of downwash will be of use. Figure 5.29 shows the two velocity components of the apparent oncoming flow superimposed on the circulation produced by the wing. The forward flow velocity produces the lift and the downwash produces the vortex drag per unit span.

Thus the lift per unit span of a finite wing (/) (or the load grading) is by the Kutta – Zhukovsky theorem:

l = pVT

the total lift being

The induced drag per unit span (dv), or the induced drag grading, again by the Kutta-Zhukovsky theorem is

dv = pwT

and by similar integration over the span

This expression for Dv shows conclusively that if w is zero all along the span then Dv is zero also. Clearly, if there is no trailing vorticity then there will be no induced drag. This condition arises when a wing is working under two-dimensional conditions, or if all sections are producing zero lift.

As a consequence of the trailing vortex system, which is produced by the basic lifting action of a (finite span) wing, the wing characteristics are considerably modi­fied, almost always adversely, from those of the equivalent two-dimensional wing of the same section. Equally, a wing with flow systems that more nearly approach the two-dimensional case will have better aerodynamic characteristics than one where

the end-effects are more dominant. It seems therefore that a wing that is large in the spanwise dimension, i. e. large aspect ratio, is a better wing – nearer the ideal – than a short span wing of the same section. It would thus appear that a wing of large aspect ratio will have better aerodynamic characteristics than one of the same section with a lower aspect ratio. For this reason, aircraft for which aerodynamic efficiency is paramount have wings of high aspect ratio. A good example is the glider. Both the man-made aircraft and those found in nature, such as the albatross, have wings with exceptionally high aspect ratios.

In general, the induced velocity also varies in the chordwise direction, as is evident from Eqn (5.31). In effect, the assumption of high aspect ratio, leading to Eqn (5.32), permits the chordwise variation to be neglected. Accordingly, the lifting character­istics of a section from a wing of high aspect ratio at a local angle of incidence a(z) are identical to those for a two-dimensional wing at an effective angle of incidence a(z) — є. Thus Prandtl’s theory shows how the two-dimensional aerofoil character­istics can be used to determine the lifting characteristics of wings of finite span. The calculation of the induced angle of incidence є now becomes the central problem. This poses certain difficulties because є depends on the circulation, which in turn is closely related to the lift per unit span. The problem therefore, is to some degree circular in nature which makes a simple direct approach to its solution impossible. The required solution procedure is described in Section 5.6.

Before passing to the general theory in Section 5.6, whereby the spanwise circula­tion distribution must be determined as part of the overall process, the much simpler inverse problem of a specified spanwise circulation distribution is considered in some detail in the next subsection. Although this is a special case it nevertheless leads to many results of practical interest. In particular, a simple quantitative result emerges that reinforces the qualitative arguments given above concerning the greater aero­dynamic efficiency of wings with high aspect ratio.

It is shown below in Section 5.5.1 how to calculate the velocity induced by the elements of the vortex sheet that notionally replace the wing. This is an essential step in the development of a general wing theory. Initially, the general case is considered. Then it is shown how the general case can be very considerably simplified in the special case of wings of high aspect ratio. The general case is then dropped, to be taken up again in Section 5.8, and the assumption of large aspect ratio is made for Section 5.6 and the remainder of the present section. Accordingly, some readers may wish to pass over the material immediately below and go directly to the alternative derivation of Eqn (5.32) given at the end of the present section.

5.5.1 Induced velocity (downwash)

Suppose that it is required to calculate the velocity induced at the point P(x, Z|) in the у = 0 plane by the L-shaped vortex element associated with the element of wing surface located at point P (.v, z) now relabelled A (Fig. 5.25).

W, S/t S*1

Making use of Eqn (5.9) it can be seen that this induced velocity is perpendicular to the у = 0 plane and can be written as

From the geometry of Fig. 5.25 the various trigonometric expressions in Eqn (5.26) can be written as

z — Z

J(x – Xi)2 + (z – Zi)2 X — XI

]{x~ Xi)2 + (z + Sz – Zj)2

/„ 7T . . z + Sz — z і

cos(02 + x) – Sin 02 = ■

y(x – Xi)2 + (z + Sz – Zi)2

The binomial expansion, i. e.

(a + b)n = d1 + ruf~lb + • • can be used to expand some of the terms, for example

where r = y{x — jq)2 + (z — zi)2. In this way, the trigonometric expressions given above can be rewritten as

Equations (5.27 to 5.29) are now substituted into Eqn (5.26), and terms involving (Sz)2 and higher powers are ignored, to give

In order to obtain the velocity induced at Pi due to all the horseshoe vortex elements, Svt is integrated over the entire wing surface projected on to the (x, z) plane. Thus using Eqn (5.30) leads to

, , 1 Ґ fx’+c dk

The induced velocity at the wing itself and in its wake is usually in a downwards direction and accordingly, is often called the downwash, w, so that w = —v;.

It would be a difficult and involved process to develop wing theory based on Eqn (5.31) in its present general form. Nowadays, similar vortex-sheet models are used by the panel methods, described in Section 5.8, to provide computationally based models of the flow around a wing, or an entire aircraft. Accordingly, a discussion of the theoretical difficulties involved in using vortex sheets to model wing flows will be postponed to Section 5.8. The remainder of the present section and Section 5.6 is devoted solely to the special case of unswept wings having high aspect ratio. This is by no means unrealistically restrictive, since aerodynamic considera­tions tend to dictate the use of wings with moderate to high aspect ratio for low-speed applications such as gliders, light aeroplanes and commuter passenger aircraft. In this special case Eqn (5.31) can be very considerably simplified.

This simplification is achieved as follows. For the purposes of determining the aerodynamic characteristics of the wing it is only necessary to evaluate the induced velocity at the wing itself. Accordingly the ranges for the variables of integration are given by – s < z < s and 0 < x < (c)maT. For high aspect ratios s/c » 1 so that x — xi| « r over most of the range of integration. Consequently, the contributions of terms (b) and (c) to the integral in Eqn (5.31) are very small compared to that of term

(a)

and can therefore be neglected. This allows Eqn (5.31) to be simplified to

is the total circulation due to all the vortex filaments passing through the wing section at z. Physically the approximate theoretical model implicit in Eqn (5.32) and (5.33) corresponds to replacing the wing by a single bound vortex having variable strength Г, the so-called lifting line (Fig. 5.26). This model, together with Eqns (5.32) and (5.33), is the basis of Prandtl’s general wing theory which is described in Section 5.6. The more involved theories based on the full version of Eqn (5.31) are usually referred to as lifting surface theories.

Equation (5.32) can also be deduced directly from the simple, less general, theor­etical model illustrated in Fig. 5.21. Consider now the influence of the trailing vortex filaments of strength <5T shed from the wing section at z in Fig. 5.21. At some other point z along the span, according to Eqn (5.11), an induced velocity equal to

will be felt in the downwards direction in the usual case of positive vortex strength. All elements of shed vorticdty along the span add their contribution to the induced velocity at z so that the total influence of the trailing system at z is given by Eqn (5.32).

In Section 4.3, it was shown that the flow around a thin wing could be regarded as a superimposition of a circulatory and a non-circulatory flow. In a similar fashion the same can be established for the flow around a thin wing. For a wing to be classified as thin the following must hold:

• The maximum thickness-to-chord ratio, usually located at mid-span, must be much less than unity.

• The camber lines of all wing sections must only deviate slightly from the corres­ponding chord-line.

• The wing may be twisted but the angles of incidence of all wing sections must remain small and the rate of change of twist must be gradual.

• The rate of change of wing taper must be gradual.

These conditions would be met for most practical wings. If they are satisfied then the velocities at any point over the wing only differ by a small amount from that of the oncoming flow.

For the thin aerofoil the non-circulatory flow corresponds to that around a symmetrical aerofoil at zero incidence. Similarly for the thin wing it corresponds to that around an untwisted wing, having the same planform shape as the actual wing, but with symmetrical sections at zero angle of incidence. Like its two-dimensional counterpart in aerofoil theory this so-called displacement (or thickness) effect makes no contribution to the lifting characteristics of the wing. The circulatory flow – the so-called lifting effect – corresponds to that around an infinitely thin, cambered and possibly twisted, plate at an angle of attack. The plate takes the same planform shape as the mid-plane of the actual wing. This circulatory part of the flow is modelled by a vortex sheet. The lifting characteristics of the wing are determined solely by this component of the flow field. Consequently, the lifting effect is of much greater practical interest than the displacement effect. Accordingly much of this chapter will be devoted to the former. First, however, the displacement effect is briefly considered.

Displacement effect

In Section 4.9, it was shown how the non-circulatory component of the flow around an aerofoil could be modelled by a distribution of sources and sinks along the chord line. Similarly, in the case of the wing, this component of the flow can be modelled by distributing sources and sinks over the entire mid-plane of the wing (Fig. 5.20). In much the same way as Eqn (4.103) was derived (referring to Fig. 5.20 for the geometric notation) it can be shown that the surface pressure coefficient at point (xi, yi) due to the thickness effect is given by

where x(z) denotes the leading edge of the wing.

In general, Eqn (5.23) is fairly cumbersome and nowadays modern computational techniques like the panel method (see Section 5.8) are used. In the special case of

wings having high aspect ratio, intuition would suggest that the flow over most of the wing behaves as if it were two-dimensional. Plainly this will not be a good approxi­mation near the wing-tips where the formation of the trailing vortices leads to highly three-dimensional flow. However, away from the wing-tip region, Eqn (5.23) reduces approximately to Eqn (4.103) and, to a good approximation, the Cp distributions obtained for symmetrical aerofoils can be used for the wing sections. For complete­ness this result is demonstrated formally immediately below. However, if this is not of interest go directly to the next section.

Change the variables in Eqn (5.23) to x = (x – xi)/c, z = z/c and z — (z – z{)/c. Now provided that the non-dimensional shape of the wing-section does not change along the span, or, at any rate, only changes very slowly St = d(yjc)/dx does not vary with z and the integral I in Eqn (5.23) becomes

1 /•! r(s-2i)/c

h = – St(x)x —

CJ0 J~{s+z)/c [X‘

, 2 72~–2

Lifting effect

To understand the fundamental concepts involved in modelling the lifting effect of a vortex sheet, consider first the simple rectangular wing depicted in Fig. 5.21. Here the vortex sheet is constructed from a collection of horseshoe vortices located in the у = 0 plane.

From Helmholtz’s second theorem (Section 5.2.1) the strength of the circulation round any section of the vortex sheet (or wing) is the sum of the strengths of the

vortex filaments cut by the section plane. As the section plane is progressively moved outwards from the centre section to the tips, fewer and fewer bound vortex filaments are left for successive sections to cut so that the circulation around the sections diminishes. In this way, the spanwise change in circulation round the wing is related to the spanwise lengths of the bound vortices. Now, as the section plane is moved outwards along the bound bundle of filaments, and as the strength of the bundle decreases, the strength of the vortex filaments so far shed must increase, as the overall strength of the system cannot diminish. Thus the change in circulation from section to section is equal to the strength of the vorticity shed between these sections.

Figure 5.21 shows a simple rectangular wing shedding a vortex trail with each pair of trailing vortex filaments completed by a spanwise bound vortex. It will be noticed that a line joining the ends of all the spanwise vortices forms a curve that, assuming each vortex is of equal strength and given a suitable scale, would be a curve of the total strengths of the bound vortices at any section plotted against the span. This curve has been plotted for clarity on a spanwise line through the centre of pressure of the wing and is a plot of (chordwise) circulation (Г) measured on a vertical ordinate, against spanwise distance from the centre-line (CL) measured on the horizontal ordinate. Thus at a section z from the centre-line sufficient hypothetical bound vortices are cut to produce a chordwise circulation around that section equal to Г. At a further section z + Sz from the centre-line the circulation has fallen to Г — 6Г, indicating that between sections z and z + 6z trailing vorticity to the strength of i5T has been shed.

If the circulation curve can be described as some function of z, f(z) say then the strength of circulation shed

6Г = – Щ^-8г (5.25)

dz

Now at any section the lift per span is given by the Kutta-Zhukovsky theorem Eqn (4.10)

l = pVT

and for a given flight speed and air density, Г is thus proportional to /. But / is the local intensity of lift or lift grading, which is either known or is the required quantity in the analysis.

The substitution of the wing by a system of bound vortices has not been rigorously justified at this stage. The idea allows a relation to be built up between the physical load distribution on the wing, which depends, as shall be shown, on the wing geometric and aerodynamic parameters, and the trailing vortex system.

Figure 5.21 illustrates two further points:

(a) It will be noticed from the leading sketch that the trailing filaments are closer together when they are shed from a rapidly diminishing or changing distribution curve. Where the filaments are closer the strength of the vorticity is greater. Near the tips, therefore, the shed vorticity is the most strong, and at the centre where the distribution curve is flattened out the shed vorticity is weak to infinitesimal.

(b) A wing infinitely long in the spanwise direction, or in two-dimensional flow, will have constant spanwise loading. The bundle will have filaments all of equal length and none will be turned back to form trailing vortices. Thus there is no trailing vorticity associated with two-dimensional wings. This is capable of deduction by a more direct process, i. e. as the wing is infinitely long in the spanwise direction the lower surface (high) and upper surface (low) pressures

cannot tend to equalize by spanwise components of velocity so that the streams of air meeting at the trailing edge after sweeping under and over the wing have no opposite spanwise motions but join up in symmetrical flow in the direction of motion. Again no trailing vorticity is formed.

A more rigorous treatment of the vortex-sheet modelling is now considered. In Section 4.3 it was shown that, without loss of accuracy, for thin aerofoils the vortices could be considered as being distributed along the chord-line, i. e. the x axis, rather than the camber line. Similarly, in the present case, the vortex sheet can be located on the (x, z) plane, rather than occupying the cambered and possibly twisted mid-surface of the wing. This procedure greatly simplifies the details of the theoretical modelling.

One of the infinitely many ways of constructing a suitable vortex-sheet model is suggested by Fig. 5.21. This method is certainly suitable for wings with a simple planform shape, e. g. a rectangular wing. Some wing shapes for which it is not at all suitable are shown in Fig. 5.22. Thus for the general case an alternative model is required. In general, it is preferable to assign an individual horseshoe vortex of strength k (x, z) per unit chord to each element of wing surface (Fig. 5.23). This method of constructing the vortex sheet leads to certain mathematical difficulties

(a) Delta wing

Fig. 5.22

Fig. 5.23 Modelling the lifting effect by a distribution of horseshoe vortex elements

Fig. 5.24 Equivalence between distributions of (a) horseshoe and (b) L-shaped vortices when calculating the induced velocity. These problems can be overcome by recom­bining the elements in the way depicted in Fig. 5.24. Here it is recognized that partial cancellation occurs for two elemental horseshoe vortices occupying adjacent span – wise positions, z and z + 6z. Accordingly, the horseshoe-vortex element can be replaced by the L-shaped vortex element shown in Fig. 5.24. Note that although this arrangement appears to violate Helmholtz’s second theorem, it is merely a math­ematically convenient way of expressing the model depicted in Fig. 5.23 which fully satisfies this theorem.

To estimate the influence of the near wake on the aerodynamic characteristics of a lifting wing it is useful to investigate the ‘hypothetical’ bound vortex in greater detail. For this the wing is replaced for the purposes of analysis by a sheet of vortex

filaments. In order to satisfy Helmholtz’s second theorem (Section 5.2.1) each fila­ment must either be part of a closed loop or form a horseshoe vortex with trailing vortex filaments running to infinity. Even with this restriction there are still infinitely many ways of arranging such vortex elements for the purposes of modelling the flow field associated with a lifting wing. For illustrative purposes consider the simple arrangement where there is a sheet of vortex filaments passing in the spanwise direction through a given wing section (Fig. 5.19). It should be noted, however, that at two, here unspecified, spanwise locations each of these filaments must be turned back to form trailing vortex filaments.

Consider the flow in the vicinity of a sheet of fluid moving irrotationally in the xy plane, Fig. 5.19. In this stylized figure the ‘sheet’ is seen to have a section curved in the xy plane and to be of thickness 8n, and the vortidty is represented by a number of vortex filaments normal to the xy plane. The circulation around the element of fluid having sides 8s, 8n is, by definition, ДГ = (8s. 8n where (is the vorticity of the fluid within the area 8s 8n.

Now for a sheet 8n -* 0 and if (is so large that the product (8n remains finite, the sheet is termed a vortex sheet of strength к = (8n. The circulation around the element can now be written

AT = k8s (5.21)

An alternative way of finding the circulation around the element is to integrate the tangential flow components. Thus

ДГ = (m2 — ui)8s (5.22)

Comparison of Eqns (5.21) and (5.22) shows that the local strength к of the vortex sheet is the tangential velocity jump through the sheet.

Alternatively, a flow situation in which the tangential velocity changes discontinu­ously in the normal direction may be mathematically represented by a vortex sheet of strength proportional to the velocity change.

The vortex sheet concept has important applications in wing theory.