Airfoils Ch10

THE FINITE WING

9.1. The Finite Wing. TTp to this point we have considered only two- tlimensional flow and have carried that development up to its current stabus as a coherent phase of aerodynamics. Now we shall turn our atkention to the finite wing and see what effect freeing the tips has on its performance.

The basic mechanism of flight was understood and practiced (with gliders) a long time ago. Many theorists became interested in its problems, and a decade before the first powered flight Lanchester (born 1868) in England postulated the type of flow that would be experienced by a finite wing. In a paper in 1894 which later led to the publication of his book "Aerodynamics"* in 1907 (Ref. 9.1) he stated that the high pressure beneath the wings would spill out around the tips into the low- pressure region above the wings, forming vortices which would stream out behind the wing. These vortices would (he continued) roll up into two main trailing vortices of opposite sign, located one behind each tip, and would be deflected do\vnward. Further, their effect upon the flow at the wing would tip the resultant force vector back, causing a component of the lift to become induced drag. It is evident that Lanchester clearly understood this phenomenon when it is noted that in 1897 he secured a patent covering the use of end plates a t the wing tips to mini- mize the spillage there-six years before the Wright brothers' flight!

Although Lanchest,er's work was mathematical in scope, his presenta- tion was not, and it remained to Pra,ndt,l (born 1875) to extend the work of Lan~hest~er into the Prandtl lifting-line theory (Ref. 9.2) presented in 191 1 and developed in tlhe following pages.

9.2. The Trailing Vortices. As Lanchester pointed out, the important difference in the flow pattern about two- and three-dimensional wings is traceable to the difference in spanwise lift distribution, which is in turn traceable to the disposition of the circulation. The two-dimensional wing has constant circulation along the span. When we represent the airfoil by a vortex, its effect ahead of the wing tending to increase the angle of attack is exactly balanced by the effect behind the wing tending to decrease it.

* Especially recommended to the student for its historical value. 181

182 BASIC W I N G AA-D AIRFOIL THEORY

On the other hand, the three-dimensional wing has a diminution of circulation toward the wing tip until, when the tip is reached, the circulation is zero. At each change of circula.tion a vortex is shed, and the shed vortices, being of similar sign, roll up into a single trailing vortex somewhat inside each wing tip. The net result is that the vortex pattern behind the wing is no longer similar to that ahead of the wing. The

4 * 4 t

from bound

2%

i I ( b ) FIG. 9.1. Downwash velocity for infinite (a ) and finite (b) wings. (The finite-wing vdue of 2wo is for a. first-approximation elliptic wing.) (Reproduced by permission from "Principles of Aerodynamics" by James H . Dwinnell, published by McGraw-Hill Book Company, Inc.. 1949.)

vertical induced velocities ahead and behind the wing no longer cancel, as the air behind the wing is affected by the downthrust on the wing circulation and the trailing vortices, while the air ahead of the wing is affected almost entirely by the upward component of the \iring circulation (see Fig. 9.1). (The effect of the trailing vortices is very small ahead of the wing.) Finally, the whole field behind the wing has a downward inclination whose vertical component is called the downwash, and at the

THE FIlVITE W I N G 183

wing enough inclination results materially to tip the resultant force backward and create a drag.

A great contribution to the theory for a complete wing was Prandtl's statement that each section of the wing acts as though it is an isolated two-dimensional section at an angle of attack a,,. This condition, known as Prandtl's hypothesis, is in general re- % sponsible for the existence of a simple theory for a complete wing of finite span. There are, however, certain conditions when the spanwise flow

7 I j

becomes so pronounced that it is not applicable, as, for instance, when severe lateral pressure gradients arise with a swept-back wing. Since these conditions become of interest in deter- mining the stalling characteristics of FIG. 9.2. Growth of shed vortices into

tip vortex, showing cancellation of com- wings, they offer an irritating obstacle ponents final reduced span.

to practical design a t the current time. They are due, of course, to the fact that curved flow ensues, owing to the above reasons, and the actual effective airfoil sections are undefined.

Another great contribution to the mathematical representation of this field was Prandtl's statement that the trailing vortices could be con-

sidered as part of one vortex which extends across the wing span and is closed a long way behind the wing by a starting vortex (Fig. 9.3 and Sect. 4.2). This complete vortex representation will enable us to calculate the induced velocity at any point near a wing as soon as

FIG. 9.3. Representation of starting "or- we develop relations to parallel in tex and a change of lift. three dimensions the simple rela-

tion of Eq. (4.3). But first a few words about this three-dimensional vortex, which now

has a finite length to the axis of rotation. Our previous vortex was flat (two-dimensional) and its core a disk. The concept of three dimensions requires a vortex of finite length and, of course, a finite-length core as well.

Helmholtz (1821-1894) in 1858 formulated vortex laws that will assist us in studying the three-dimensional vortex as well as the vortex pattern of a finite wing. Briefly these were:

184 BASIC JVIA-G AIVB A I R F O I L T H E O R ) .

1. Vortex filaments either form closed curves or extend to the fluid boundaries.

2. The circulation remains constant throughout the length of a vortex. 3. The particles of fluid composing a vortex remain with that vortex

indefinitely. Sir William Thomson (Lord Kelvin, 1824-1907) is usually credited

with the additional law, "Circulation remains constant with time." Let us see how well actual conditions meet these la\vs. Flow pictures made (Ref. 9.4) a t the instant air begins to move over a

wing a t some lifting angle of attack show that a vortex, parallel to the wing span, leaves the wing trailing edge and proceeds downstream. This vortex is called the starting vortex and is of strength equal to that of the lifting, or bound, vortex. The sum of the t ~ v o is thus zero, satisfying the Thomson law.

The starting and bound vortices may be very simply demonstrated with a razor blade and a basin of water. The razor blade, held half immersed so that i t represents a wing a t some lifting angle, is moved briskly through the water. A starting vortex immediately leaves the blade. As long as the motion is constant, no additional vortices appear, but any increase of angle produces a second starting vortex. When the motion is stopped, the bound vortex comes off the "wing." The starting vortex, by connecting the ends of the two trailing vortices of the wing vortex system, enables that system to meet the first Helmholtz law. The system meets the second law by actually having constant circulation in each vortex section. The third law is not quite satisfied, since the system is continually picking up air and thrusting it into the trailing vortex system. Indeed, in view of the molecular action taking place in any gas there is doubtless a considerable exchange of molecules between those in the core and in free air.

After the starting vortex has moved downstl-eam to a point where its influence near the wing is negligible, there remaifis only the bound vortex with two trailing vortices in the region. The shape of this combination resembles a horseshoe, and ordinarily the system is so called, instead of identifying its components.

Since the lift vector of a circular cylinder goes through its axis, the horseshoe vortex system is most accurate if the bound vortex is assumed to run through the wing center-of-pressure line.

We see directly that our simple horseshoe vortex field, having its vortex strength determined from L = pVI'b, fits in well with the actual conditions. Silverstein, Katzoff, and Bullivant (Ref. 11 .l) examined this phenomenon by comparing the flow field about a Clark Y as determined by the method of Chap. 8 and the vortex lams of Chap. 4. It was

THE FINITE WING 185

discovered that a t a distance of one chord from the wing trailing edge the downwash angles were matched within 0.3" by a simple point vortex field.

While we have just discussed the case of constant circulation across the span, i t may be seen by reference to Fig. 9.4 that any symmetrical (or, indeed, unsymmetrjcal) spanwise distribution of lift may be simu- lated by a number of superimposed closed vortex systems. In actuality, these small vortices, shed a t every change of circulation on the span, roll up into the tip vortices and move them slight,ly toward the plane of

- - - ---, / distribution Actual lift

C -

(a) FIG. 9.4. Theoretical vortex pattern for arbitrary distribution of circulation along the span. roll-up of shed vortices neglected. (Reproduced hy permission from "Principles of Aero- dynamics" by James H. Dwinnell, published by McGraw-Hill Book Company, Inc., 1949.)

symmetry. In addition the whole field aft of the wing is displaced vertically downward by the bound vortex.

The manner in which the circulation varies along the span is of great interest to the aerodynamicist since it will be shown that one particular variation (elliptic) results in minimum induced drag, others producing from 2 to 15 per cent more. All variations of the circulation are amena- ble to mathematical treatment, but for simplicity we shall consider four: elliptic (produced by an untwisted wing with elliptic planform; uniform (not actually produced by any planform, but of such simple mathematical scope that i t helps in understanding the other cases); rectangular-wing type; and tapered-wing type. I t is emphasized that rectangular wings do not have uniform loading, for reasons that will be brought out later.

186 BASIC WllQG AND AIRFOIL TIfEORY

9.3. The Biot-Savart Law. Before \ye can actually calculate the effect of the trailing vortices, it is necessary to develop vortex laws for three-dimensional flow. Added value may be attached to this particular development as i t embraces nearly every aerodynamic and vector princi- ple studied so far.

First let us recall some of the rules and definitions of the velocity potential. 4 has been defined by t,he relation

\\-here the symbols are shown in Fig. 4.11. The velocity potential of one point A relative to another point A' is

I t was demonstrated in Sect. 4.6 that 4 exists only for irrotational motion and is then independent of t,he path of integration-in this case AA' in Fig. 9.5. It is further noted in Chap. 4 that the value of the circulation bears a close resem- blance to Eq. (9.3): u- Aa r = $ q . d s (9.4)

The only difference is that for A circulation the path of integration

FIG. 9.5. is closed. Under these conditions, if the line integral about the closed

curve is not zero, its value is the circulation, and the motion is not completely irrotational unless a solid body is included.

Now consider an arbitrary surface which has a vortex for a boundary. The line integral along any closed path 1M that does not cut the surface is zero. But when the surface is cut, we get a sudden rise in potential due to the vortex. It is advantageous to assume a thickness to the surface to avoid an infinite rate of rise (see Fig. 9.6).

Now a given potential may be produced by sources and sinks or by vortices. At this point it is more convenient to consider the flow field produced by the closed vortex line of Fig. 9.5 as being produced by a double sheet of sources and sinks as shown in Fig. 9.7. For instance, if Q is the source quantity, then the velocity at point P a distance r

THE FINITE WING 187

away will be

q = B 4nr2 (9.5)

[Compare Eq. (9.5) with the development of Eq. (3.13).]

Potential

The potential a t point a due to a source at the origin is

which in this case is

sheet

But q, is straight (the flow from a source is radial), and the angle be- ! tween q, and dr is zero (Fig. 9.8), so that,

,, [ qr dr 4 ==

(9.8) - / O x d r = -- 0 42w2 47ra

or for a sink Q 4a = - 4na

If we call the source intensity Q1, the potential at P due to elemental FIG. 9.8.

area d A of the doublet is

188 BASIC WING AND AIRFOIL THEORY

where a, and as are the distances from P to the proper source and sink elements.

We see, however (Fig. 9.9), that

where hl is the distance between source and sink sheets, and hence

Excluding the last term of Eq. (9.1 1) as small, we have

We may drop the subscript on the a2 term as with hl small and a >> hl, ai g aa, and in the term where a2 appears the difference between ai and a, is inconsequential.

Equation (9.9) is then

Now consider the flow between upper and lower sheets. Except for a very small (but important) amount, the entire flow is from source to sink between the sheets. For a square-foot area the quantity is Q1 cu f t per sec, and hence the velocity developed from upper to lower sheet is numerically equal to Q1 f t per sec. 4 s this takes place along a distance hl, we see the rise in potential = velocity X distance = Qlhl.

As we have defined the flow as producing a rise in potential equal to bhe vortex strength (or I?), then Qlhl = I' and Eq. (9.13) becomes

Equation (9.14) relates the velocity potential with the circulation but is not yet in a usable form. Indeed, from a practical standpoint, we shall want to change it to another general form.

Now, return t,o the vortex-sheet concept, and put a small unit spherical surface about the point P (Fig. 9.10a). From Fig. 9.10b the projection of d A perpendicular to a is essentially d A cos a, and, assuming d A cos a

to be square, the rays from d A to P will define a small "square" (or

THE FINITE WING 189

solid angle w ) on the surface of the sphere. On the unit sphere the projection will be inversely proportional to the square of the distance, or

dA cos a dw = - a2

From Eq. (9.14)

or for the whole surface bounded by the vortex

Now the velocity a t point P is

We seek q due to the vortex of strength I', and our development so far tells us that q is a function of the solid angle w. I t is apparent that w

would change if the point P were moved from the surface or, equally, if the vortex surface were moved from the point. The latter approach, it develops, is easiest to analyze. We must move the sheet along a, hold-

190 BASIC JPI'lrC A S D AIRFOIL THEOR)

ing it parallel to itself, as any angular motion vould correspond to rotating the point P about the sheet, and v e n-ish to examine the effect of only moving P.

From inspection, increasing the distance a decreases the potential a t P as it decreases w. Thus it is that 17:hile we consider the effect of moving the vortex sheet we are also considering the effect on the potential at P.

From Fig. 9.11, the only movement of the vortex sheet that affects w

will be that which affect,s the projected area dA1 = d A cos a. Moving

the sheet parallel to itself, we see the net result is a decrease of o by a perimetric strip, and we may evaluate this strip as follo~vs: The displace- ment along a of the vortex sheet parallel to itself creates a surface of height dr and perimeter $ ds. A section of this surface is ds dr, and the projection of ds dr on the perpendicular to a is ds dr cot 02, where O2 is the angle between the radius vector PL and a perpendicular to a, BN.

Now dr is perpendicular to ds, and the vector that represents the arca ds dr may hence be written e ds dr or ds x dr. The vector that represents the area dAl will be perpendicular to dAl and hence lie along a and have a unit length al = a/a. In addition the angle O2 will be nearly 90°, and we may write cot O2 = cos 02. Finally the change in projected area as the sheet is moved becomes

THE FINITE WING

and the change in the solid angle w is then

a3 (9.20)

which by Eq. (1.25) becomes

As we have moved the vortex sheet parallel to itself only, the direction and length of dr is constant. Hence we may remove dr from the integral [Eq. (9.21)] as follows:

du = dr fdT, (9.22)

To digress a moment,

From the definition of a derivative we then have

dr V w = dw So from Eq. (9.23)

d r - V w = d r - f y

and

Substituting in Eq. (9.18),

and substituting for d s x a its equal, ca ds sin 0, where 0 is the angle between a and the vortex segment ds, we have (see Fig. 9.126)

ds sin 9 191 = &/, ,-

Now, having the expression for the effect of a segment of vortex, we may proceed to the application problems. First, however, it is convenient to express Eq. (9.25) in trigonometric terms.

Taking out a section a dB (Fig. 9.12a), we see that d x = a dB;

sin B = dx/ds so that

a dB d ~ = - sin 0


Also from Fig. 9.12b sin 0 = h/a, so that

Changing limits, when s = A, 0 = 01 and when s = B, 0 = 02, and, substituting, Eq. (9.25) becomes

I ' lql = - (cos 02 - cos 01) 4rh

Considering the two interior angles as easier to use, we have

Equation (9.27) is a common form of the Biot-Savart law. I t demonstrates first of all that, as the vortex segment is approached, the induced

velocity i t produces rapidly increases. Now, according to the Helmholtz laws the vortex cannot abruptly end in the fluid, but when i t turns a sharp corner the usual approach is to consider each straight segment separately, just as though it did terminate at the corner. Thus for a vortex that "starts" at P and extends to infinity out beyond B, el = r j2 , and 83 = 0, and

r 9 " q,h. (9.28)

Or if the vortex extends to infinity in both directions, el = e3 = 0 and

Of interest is the fact that the induced velocity exists beyond the "end" of a vortex, although its value is small (Fig. 9.13).

The replacing of a wing by a simple lifting-line vortex and the proper use of the Biot-Savart law are illustrated in subsequent chapters.

THE FINITE WING 193

9.4. Downwash at the Wing for Any Spanwise Distribution of Circu- lation. Undoubtedly the most important effect of the trailing vortices

FIG. 9.13. Values of induced velocity near the "end" of a vortex segment, I' = 4 0 0 ~ sq It per sec.

is the increase of drag they produce through altering the flow angle a t the wing. Secondarily, the manner in which they alter the flow at the wing is important, too, as usually the induced velocity is not constant across the span, and the local angle of attack varies over a large range.

Let us call the downwash velocity w and the wing span b. Orient the wing so that the x axis coin- cides with the forward airplane

+-- ------- longitudinal axis, the y axis coin- 1:

cides with the bound, or lifting, vortex, and the z axis is downward. (Downwash is positive in the downward direction.)

Since the circulation decreases to- Yz Fro. 9.14.

ward the tips (it must be zero at the tip), between the points y and y + dy on the span the circulation changes

ar - - dy. Hence the induced velocity a t the point yl for a differential a~ vortex a t y is, from Eq. (9.28) and Fig. 9.14,

and the total downwash a t yl will be

* dw(yt) should be read "dw at yl."


Equation (9.31) hence defines the amount of downwash that will occur at point yl for any arbitrary distribution of circulation described by

-. dr I t is known as the Prandtl downwash relation. 8~

Using the local down~vash w, ~i-e may then find the local angle of attack and local lift and finally the approximate span load distribution. Repeating the process, we could in theory find the actual span loading by eliminating the effect of each new downwash distribution on the original assumed span loading. However, the problem is extremely complex, and we shall usually confine ourselves to considering a uniform or an elliptic span loading in this and the following chapter. (In Chap. 12 two methods for getting the span loading for arbitrary planforms are discussed.)

9.5. Downwash at the Wing for Elliptic Distribution of Circulation. Before proceeding further it is in order to discuss the terms span loading, spanwise lift distribution, and spanwise lift coeficient distribution. Since

[Eq. (4.4)] the lift is a function only of the density, velocity, and circulation, for a given set of conditions an elliptic distribution of circulation would also be an elliptic distribution of lift. However, the relation between the section lift and

no. 9.15. the section lift coefficient involves the local chord, and hence elliptic

lift or distribution of circulation may or may not mean elliptic distribution of cl. In fact, the only case where all three distributions are elliptic is that of the properly twisted rectangular wing.

Proceeding now to the case of elliptic loading ("loading" as we have seen may be either lift or circulation), we assume a wing that has a circulation ro at mid-span which decreases elliptically until it is zero a t the wing tips (see Fig. 9.15). We write

or

Then

T H E F I N I T E W I N G 195

Substituting into Eq. (9.31), we get that the downwash a t point yl on the span is

The integrations of Eq. (9.35) is complicated by the fact that the value of the integrand is infinite when y = yl, b/2 or -b/2. The complete integration is given in Appendix 3, and we may hence state

But yl is any point on the span, and hence this relation applies to every point on the span.

The interesting conclusion'is that, if the distribution of circulation is elliptic across the span, the downwash a t the wing due to the trailing vortices is constant across the span. While we might rightfully con- clude that the local angle of attack is also constant for an untwisted elliptic wing, the conclusion cannot be extended to include a simultaneous stall across the span, as practical considerations of viscosity dictate an earlier stall for the shorter chord, lower Reynolds number wing tips.

9.6. Downwash at the Wing for Uniform Distribution of Circulation. The case of uniform distribution of circulation is developed in Chap. 11, but we may state the results here to compare with Sect. 9.4. It is shown in Chap. 11 that the downwash distribution a t a wing with uniform loading is

(The symbols are defined in Chap. 11.) Substituting for example r = 6.28 and s = 1.0, we get the values of w

shown in Fig. 9.16. Here it is seen that the angle of attack is locally reduced by uniform loading all along the span, but unlike the case of elliptic loading the reduction is not uniform, being greatest at the tips. This tends toward a wing root stall appearing earliest. We may also get nearly uniform loading by twisting a tapered wing, but in practice the twist is kept small to avoid excessive deviations from the elliptic- type loading, for reasons to be advanced later.

As a matter of interest the effect of the trailing vortices on the uni- formly loaded wing quite obviously tends toward destroying its uniformity of lift, and hence even the rectangular wing tends toward an

BASIC WING AND AIRFOIL THEORI'

-M

THE FINITE WING 197

elliptic distribution. Experience tells us it gets about halfway there, but the alteration from the basic uniformity is not enough to alter the expected root stall.

An interesting check of this theory and of Fig. 9.16 was made by Piercy and is reported in Ref. 9.3. In these experiments a wing of 12 in. span and 3 in. chord was set at 8" in a wind tunnel with a 4-ft-square test section. Using a small yawhead, the angle of flow was measured 0.9b behind the wing in a plane near that of the wing. The results, shown in Fig. 9.17, demonstrate clearly the existence of a vortex similar to that supposed by the theory. At a distance of 0 . 5 ~ behind the trailing edge a trailing vortex was clearly discerned. I ts core had a diameter of 0 . 2 ~ ~ and its center was 0.95s from the plane of symmetry. Inside the core the linear velocity appeared very nearly proportional to the radius (in accordance with Fig. 4.3), and outside the core the velocity closely checked the values of Eq. (9.37). The vortex span was somewhat greater than predicted in Sect. 11.6 but remained constant with a.

The diameter of the vortex core (for which no satisfactory theory has been devised) increased slightly with the circulation. The center of the core for this particular case revolved a t over 18,000 rpm.

The high speed of the vortex core produces interesting and beautiful effects in air under special conditions. The high speed implies a low pressure, and for air we then get a low temperature. Since the amount of moisture the air can hold decreases with falling temperature, vortices in humid air becomes visible, appearing somewhat like steam in air, as the moisture condenses out. This phenomenon is a familiar sight a t the wing tips and landing-flap terminations and, more frequently, streaming helically behind propeller blade tips a t take-off.

9.7. Induced Drag Coefficient. We may now examine the effect of the downwash on drag for the simplest case-that of elliptic loading.

For the finite wing, Eq. (4.25) becomes

and for elliptic loading in accordance with Eq. (9.33) we get

Making the substitution that y = (b/2) sin 0, dy = (b/2) cos e de and finding new limits we have


Integrating and solving for Fo, we have

Substituting into Eq. (9.36),

and it is seen that the effect of downwash is to deflect the airstream through an angle

W a, = - v (9.43)

as is shown in Fig. 9.18. The changes in lift and velocity are small and may be neglected, but the effect on drag may be large.

Considering the dragwise compo-

L nent of the lift Di, we have

D, = L tan a; = L - (9.44) (3 and from Eq. (9.43) this becomes

D, = L2

(p/2) V 2 rb2 (9.45)

FIG. 9.18. Multiplying both sides of the equation by V and dividing by 550 to get horsepower we have, since the lift must equal the weight W in steady unaccelerated flight,

where q = dynamic pressure (p/2)V2. Putting V in miles per hour and letting a = p/po, where po = density at standard conditions, the final form is

Equation (9.47) demonstrates that the horsepower required to over- come induced drag is a function of the s p a n loading, the velocity, and the density and gives a clearer insight into the problem than the usual

CL* CDi = - T A.R.

T H E FINITE W I N G 199

which may be found from Eq. (9.45). Equation (9.48) seems to indi- cate that induced drag is a function of aspect ratio, where, more accu- rately, only the drag coefiient varies with aspect ratio, and the area changes that occur with changing aspect ratio and constant span are such that. the induced drag remains constant with constant weight and speed.

It should be noted that Eqs. (9.45), (9.47), and (9.48) were derived for elliptic loading only.

PROBLEMS

9.1. If a wing of 6 f t chord has a local lift coefficient of 0.6 and is operating in a 100-mph airstream, what is the local circulation?

9.2. Prove that the untwisted elliptic wing gives elliptic distribution of circulation.

9.3. What is the downwash in feet per second at the wing centerline of a wing with uniform loading if its lift is 10,000 Ib, span 30 ft, V = 200 mph, and the air standard sea-level density?

9.4. Find the induced velocity 4 ft from a doubly infinite vortex of strength r = 30 sq f t per sec.

REFERENCES

9.1. F. W. Lanchester, "Aerodynamics," Constable & Co., Ltd., London, 1918. 9.2. L. Prandtl, Applications of Modern Hydrodynamics to Aeronautics, TR 116,

1921. 9.3. N. A. V. Piercy, On the Vortex Pair Quickly Formed by Some Airfoils,

JRAS, October, 1923, p. 489. 9.4. S. Goldstein, Modern Developments in Fluid Dynamics, Vol. I, p. 69,

Oxford University Press, New York, 1938. 9.5. N. A. V. Piercy, "Aerodynamics," p. 235, English University Press, London.

Documents

Airfoils Ch10