Two Practical Conversion Formulas

The formula given earlier linking the induced drag CDi to the lift CL led Prandtl to two further results that were significant for engineering and indicative of the power of the analysis he was developing. He discovered two practical formulas that allowed him to reduce drastically the amount of time spent conducting wind-channel tests. It allowed him to generalize results about in­duced drag and induced angle of incidence from a wing that had been tested to other wings with a different aspect ratio that had not been tested. All that was necessary was that the wings have the same cross-sectional profile. In this section I explain how Prandtl was able to do this.

The formula (deduced above) for the coefficient of induced drag, namely,

is analogous to the equation y = ax2, where y = CDi and x = CL, while a is the constant i/kAr. Equations having the form y = ax2 generate curves in the shape of a parabola. The constant a determines the curvature of the pa­rabola. According to Prandtl’s theory, then, the curve linking the coefficients of induced drag and lift will be parabolic, and the curvature will depend on the aspect ratio. Is this theoretical deduction confirmed experimentally? The “polar curves,” as Lilienthal called them, which link the observed coefficients of lift and drag are, indeed, roughly parabolic, but some distinctions need to be drawn before such results can be related to the parabolic formula for the induced drag.

Induced drag arises in an ideal fluid but also plays a role in real fluids. There are, however, other sources of drag that are present in real fluids. A real wing in real air will experience some degree of skin friction due to the viscos­ity of the air. As Betz had previously argued, the flow of air over a real wing will also generate eddies, and the flow over the upper and lower surfaces will not join together smoothly at the trailing edge. This, too, is a source of drag. Prandtl grouped both of these latter sources under the name “profile drag.” Thus the empirical phenomenon of drag is the combined effect of induced drag and profile drag. The empirical coefficient of drag can be understood as the sum of the coefficient of induced drag and the coefficient of profile drag. Thus, using obvious notation,

CD = CDi + CDp.

If the theoretical polar curve relating induced drag and lift is plotted on the same diagram as an empirical curve relating total drag to lift, it turns out that the curves are similar but not identical. They lie close to one another but do not overlap. Their relative position shows that most of the drag on the wing is induced drag. This is particularly true at high angles of incidence (see fig. 7.13).

Prandtl realized that the contingent relation between the two drag curves could be exploited. He spoke of a “fortunate circumstance that had not been suspected at all at the outset”—“einen glucklichen Umstand, der von vorn – herein keineswegs vermutet wurde.”75 It was a stroke of good fortune that al­lowed the results of his wing theory to be cast into a form that was highly use­ful for practical purposes (“gelang es nun, die Ergebnise der Tragflugeltheorie auch fur die Praxis noch wesentlich fruchtbarer zu gestalten”). As Prandtl explained:

Man trug namlich die Polarkurven bei sonst gleichen Tragflugeln, aber von verschiedenen Seitenverhaltnissen auf und erkannte, dafi der Unterschied der gemessenen Widerstandszahl von der theoretischen in allen Fallen nahezu der gleiche war. Daraus war zu schliefien, dafi der Profilwiderstand unab – hangig vom Seitenverhaltnis ist, woraus sich weiterhin die Moglichkeit ergab, gemessene Polarkurven von einem Seitenverhaltnis auf ein anderes Seitenver­haltnis umzurechnen. (219-20)

The polar curves of wings that are the same, apart from having different aspect ratios, were laid out, and one could see that the difference between the mea­sured and the theoretical resistance coefficient was approximately the same in all cases. It had to be concluded from this that the profile resistance was independent of the aspect ratio. This raised the possibility that the measured polar curve for one aspect ratio could be converted into that for another as­pect ratio.

Prandtl’s compressed reasoning can be broken down into two steps. First, the total drag is the sum of induced drag and profile drag (and the larger and more important of these two quantities is the induced drag). Second, if the profile drag CDp is roughly the same for wings of all aspect ratios, then the difference between total and induced drag would be the same if the wings were operating at the same coefficient of lift. In other words, the quantity ( CD – CD) would be a constant. This quantity could therefore be equated for wings of different aspect ratio, provided they both had the same profile and both had an elliptical lift distribution. The implication was that given two wings, wing(1) and wing(2), the total drag of wing(2) could be predicted once

figure 7.13. Two polar curves relating induced drag and lift. The curve at the front on the left is derived from theory, the other is plotted from measurements. From Tietjens 1931, 219. (By permission of Springer Science and Business Media)

it was known for wing(1). If the aspect ratios and the coefficients of total and induced drag for the respective wings are also distinguished by the labels (1) and (2), Prandtl was able to write

Q>(1) – CDI.(1) = Cd(2) – CDi(2).

It had already been established that

so this result could be substituted into the above equation to give

Rearranging the equation gives the drag of one wing in terms of the drag of the other, at the same value of the lift coefficient. The formula thus allows knowledge about one wing to be converted into knowledge about the other wing. The conversion formula was thus

П Li___ L_ I.

П і A (2) Ar (1))

A second conversion formula was then deduced. This formula dealt not with drag but with the angle of incidence. Once again it converted knowledge gained from one case into knowledge applicable to other cases. The second formula implied that if the angle of incidence associated with a given lift is known for a wing of one aspect ratio, then the angle at which that lift was pro­duced could be predicted for another wing of the same profile but a differ­ent aspect ratio. In this case the reasoning depended on the relation between finite wings and an infinite wing with the same profile.

Prandtl argued as follows. Suppose an infinite wing, of a given profile, meets a horizontal airstream at an angle a0. Let the lift coefficient be Cl. What would happen if this profile were to meet the air at the same speed but now as part of a finite wing, not an infinite wing? Prandtl had shown that the ef­fect of the vortices, which now trail from the tips, is to induce a downward flow of the air that presents itself to the wing. This induced angle ф reduces the effective angle of incidence of the wing. If the finite wing is to generate the same amount of lift per unit length as the infinite wing, then it must be restored to the same angle relative to the local flow that it originally had to the free stream. This can only be done if the angle to the horizontal is increased.

The angle of attack a will have to be made equal to the original angle a0 plus the induced angle of incidence ф. Thus, a = a0 + ф. Rearranging the equa­tion leads to a0 = a – ф. This expression implies that, for all wings of the same profile, the difference between the angle of attack and the angle of the induced flow will be the same when they are delivering the same amount of lift. Thus for two such wings, using obvious notation,

a0 = a1 – ф1 = a2 – ф2.

Suppose that wing(1) has the lift coefficient CL at a1 and wing(2) has the same lift coefficient at a2. Prandtl had already arrived at an expression for the in­duced angle of incidence ф, so he could write

C. 1 C. 1

a. = a. — —- = a — —- .

0 1 n Ar (1) 2 n Ar (2)

This gives the second conversion formula

Cl I 1 1 I

a=a +—I———— .

2 1 4 Ar (2) AR (1) J

If these two formulas stood up to test, they would fulfill the desiderata for work in technical mechanics identified by August Foppl. I described in chap­ter 5 how Foppl had insisted that the role of time in the economy of knowl­edge was different for the engineer compared to the physicist. The value of the conversion formulas was that they would enormously lighten the work load of the engineer engaged in wind-channel research.

The formulas were first published by Betz in 1917 in the confidential Tech – nische Berichte.76 They were then tested in Gottingen by taking wings of dif­ferent aspect ratios in order to see if the measurements for drag and angle of incidence could be converted into the values for one, arbitrarily chosen, aspect ratio. The first such test was performed by Munk and was also re­ported in the Technische Berichte71 Munk used just three different aspect ra­tios and, in order to keep the section of the wings as constant as possible, simply started with a long span of wing and sawed off the ends to produce the shorter wings. In this way he produced wings of aspect ratio 6, 5, and 4. Munk verified the formulas by calculating the results for the wing of aspect ratio 6 from the other two aerofoils and plotting the three sets of points in the same graph. Later the experiment was repeated with seven different aspect ratios and produced the same positive result. Except for the measurements taken on one wing of very low aspect ratio, the predictions worked well. The conversion formulas did what they were meant to do, that is, collapse all the experimental results into one and the same curve.78

Any direct test of the conversion formulas was also an indirect test of the theoretical assumptions on which they were based. As well as sanction­ing a practical shortcut that avoided much time-consuming work with the wind channel, the positive results of the test were a corroboration of Prandtl’s overall analysis. But all of the reasoning rested on the assumption that the lift distribution was elliptical. This facilitated the calculations but made the result a special case. Could the result be generalized? In November 1913 Prandtl and E. Pohlhausen had established that the induced drag for an el­liptical lift distribution was not only constant along the span but represented a minimum value.79 Any deviation from an elliptical distribution would give a higher value for this form of drag. It was also soon established that the actual planform that produced an elliptical lift distribution was itself of an elliptical shape. This was not because there is any simple rule to the effect that wings generate distributions that mirror the shape of their planform. In general, the shape of a wing does not immediately correspond to that of the resulting lift distribution. A rectangular wing does not yield a rectangular lift distribution. But, despite having the character of special case, it turned out that all the results derived for the elliptical wing could be generalized. The mathematical apparatus that has just been sketched could be applied, with­out significant loss of accuracy, to non-elliptical wings, for example, to the simple-to-construct rectangular wing that was used as the baseline or “norm” (the Normalflugmodell) in the Gottingen profile tests.80

The empirical basis for the generalization has already been mentioned. It rests on the “family resemblance” between the lift distributions of all typi­cal wings. Though their lift distributions are not strictly elliptical, they are, mostly, roughly elliptical. As Prandtl pointed out, while the true ellipse gives the minimum possible induced drag, many mathematical functions change their values slowly in the vicinity of a minimum. Results that hold for the minimum are often found to hold, at least approximately, in the neighbor­hood of the minimum.81 Thus the Gottingen results had a practical applica­bility, and a predictive power, that went beyond what might have been ex­pected, given the specialized, and often unreal, assumptions on which they were based. Looking back, some quarter of a century after the creation of Prandtl’s theory, Richard von Mises summed up the situation as follows: “It seems appropriate to stress the fact that. . . the parabolic form of the polar diagram and the dependence of this form on the aspect ratio, and the relation between lift coefficient, angle of attack, and aspect ratio, were not known as empirical facts before the wing theory was developed. These facts . . . have been predicted by the theory. Experiments carried out a posteriori have con­firmed these theoretical predictions to a degree that is remarkable in view of the numerous idealisations of the theory.”82 Von Mises was not a wholly unqualified admirer of the Gottingen group—he thought they cited one an­other too much—but the word “remarkable,” applied to the success of the Gottingen theory, was reiterated in his book Theory of Flight.83 The repetition attests to the striking and, it would seem, almost baffling power of Prandtl’s work. Let us look a little more closely at some of the methodological features that were associated with this success.