Glauert Makes the Case

On March 30, 1920, before he went to Germany, Glauert had presented his “Notes on the German Aerofoil Theory” to a meeting of the Aerodynam­ics Sub-Committee which included Glazebrook, Greenhill, and Lamb. The notes amounted to a brief overview and assessment of two conversion for­mulas. Part 1 of the paper dealt with the formula linking the aerodynamic characteristics of monoplane wings of different aspect ratios, while part 2 concerned the link between monoplane wings and biplane configurations. Glauert stated the relevant formulas without proof and simply said that they were taken from “German Technical Reports.” In his comments to the sub­committee, when introducing the notes, he added that he had not yet been able to locate the papers giving the theory on which the formulas were based. His aim was to marshal some empirical data to find out if the formulas gave the right answers. He concluded that in some cases they did but in some cases they did not. In general, the transformation formulas discussed in part 2 of the notes seemed problematic, while those in part 1 worked well for predict­ing the induced drag but badly for predicting the induced angle of incidence. Because the (good) result for drag is theoretically dependent on the (bad) result for angle of incidence, Glauert declared himself puzzled.

These notes provided the basis for an article that Glauert published soon afterward in the short-lived journal Aircraft Engineering.66 The article gives further insight into the status he accorded to Prandtl’s theory before the Got­tingen visit. Glauert put it like this: “Good agreement is not obtained for the angle of incidence, and as the theory estimates the change in drag from the effective change in incidence, it is evident that the basis of the theory cannot be regarded as quite satisfactory. The form of the expression found for the induced drag has a certain theoretical justification, but it is probably safer to regard the results as empirical formulae which are confirmed by experi­mental results” (161). The tone of this conclusion, in which Prandtl’s results were accorded the status of mere “empirical formulae,” contrasted with that adopted after his visit with McKinnon Wood to Gottingen and his talk to Prandtl.

In February 1921, after his return from Gottingen, Glauert produced his report T. 1563 on the outcome of his talk with Prandtl. The report was titled simply “Aerofoil Theory” and was based on six sources, all by Betz, Munk, and Prandtl.67 These sources included Prandtl’s Tragflugeltheorie and the Gottingen dissertations of his two assistants. Glauert’s aim was to give “an account of the development of the theory and of the main results contained in the original papers” (2). He divided his report into five sections: (1) aero­foils of infinite span, (2) the finite monoplane wing, (3) special cases of the monoplane wing, (4) biplane wing structures, and (5) the influence of walls and the free boundaries of a stream on the flow in a wind channel. What fol­lowed was one of the most lucid accounts that has ever been given of the ba­sics of the subject. Farren and Tizard refer to the “faultless style” of Glauert’s exposition.68 Although some of the same reservations were carried over from the earlier “Notes on the German Aerofoil Theory,” for example, the empiri­cal weakness of the prediction of the induced angles of incidence, these were not deemed to be of great practical importance compared to the accurate predictions of induced drag. Furthermore, the fuller treatment of the relation between monoplanes and biplanes had removed some of the earlier doubts. In the light of further analysis, Glauert now concluded that “the theoretical formulae may be accepted as giving a reasonably accurate method of predict­ing the biplane characteristics from those of the monoplane” (26).

How was Glauert’s report received? What, for example, did the Aerody­namics Sub-Committee make of it? At meeting 38 of the subcommittee on April 5, 1921, minute 375(b) records that “Prof. Lamb remarked that he had read the report with great interest and considered it a very valuable addition to aerodynamic theory.” Lamb did, however, say that he found the vortex lines difficult to visualize, and J. D. North suggested that the relevant dia­grams were to be found in Lanchester’s book.69 (Whether Lamb found those diagrams acceptable or whether, like Prandtl, he thought they were wrong, is not recorded.) Although one may wonder about the identity of the (implied) prior theory, to which Prandtl’s theory was a “valuable addition,” Lamb’s re­sponse may seem positive enough. There is, however, a second version of Lamb’s reaction which must put a question mark over this positive inter­pretation. The second version is given in the minutes of the full Aeronauti­cal Research Committee that met for its tenth meeting a few days later, on Tuesday, April 12, 1921, at the Royal Society. (Lamb now served on both the Aerodynamics Sub-Committee and the full research committee.) Minute 111 of the full committee meeting deals with the business of the subcommittee and refers to Glauert’s “Aerofoil Theory” as “report (ii).” It reads as follows: “The report (ii) was stated by Professor Lamb to form a good basis for the commencement of work on the development of an aerofoil theory. Professor Bairstow expressed his dissent.”

Had Lamb moderated an earlier, more positive response or did the later minutes simply capture nuances that were lost in the earlier summary? When one recalls the highly qualified wording that Lamb had used in his Hydrody­namics, when describing Kutta’s work, the later minute seems closer to the authentic voice of this cautious spokesman of the Cambridge school. Either way, the full Aeronautical Research Committee did not receive Glauert’s ac­count of the Gottingen work with open arms. Bairstow was clearly not im­pressed by what he was hearing of Prandtl’s achievements, and Lamb’s ap­parent support now had so many qualifications that it is difficult to decide whether he was really being supportive or not. To say that something is a “basis” for a “commencement” of a “development” is not to say a great deal.

Undeterred by this response Glauert presented a second report in May 1921 called “Some Applications of the Vortex Theory of Aerofoils,” which dealt with both wing theory and propeller theory. (I confine myself to the former.) Glauert was clearly in no mood to compromise and began by assert­ing that his previous paper had led to “a satisfactory theory for correlating the lift and drag of different wing structures and for determining the effect of changes of aspect ratio.” His aim now was to see whether it gave an accurate picture of the flow of air in the vicinity of the wing. Glauert’s talk of the the­ory “correlating” data suggests he may have still been concerned lest Prandtl’s approach merely provided empirical formulas rather than a physically true account of the actual air flow. His intention was to address this anxiety by comparing the calculated and observed “downwash” of air at three locations in the vicinity of a wing: (1) above or below the center of the wing, (2) behind the (main) wing in the region of the tailplane, and (3) at the wingtips.70

Before making the comparison Glauert entered a caveat. Prandtl’s theory rested on drastic simplifications, and these would necessarily preclude it giv­ing an accurate picture of certain features of the flow. First, the wing was replaced by the abstraction of a “lifting line.” For both the simple horseshoe model and the refined model, with a varied distribution of circulation along the span, the chord of the wing was neglected. So the flow close to the wing could not possibly be described accurately. Second, where a vortex sheet was assumed to be issuing from the trailing edge, the sheet would roll up, so the flow behind the wing would have a different character at different distances. As a partial response to this second problem Glauert performed his calcu­lations of the downwash in two different ways: (1) on the assumption of a constant distribution of lift (the simple horseshoe model) and (2) on the as­sumption of an elliptical distribution of lift (the refined horseshoe model). He argued that the rectangular wing used in the experimental tests would have a lift distribution somewhere between these two extremes. Further­more, the trailing-vortex system near the wing would be more like the re­fined model, whereas the system at a distance would be more like the simpler model. Glauert argued that provided the tests were not carried out too near the wing, or too far behind it, the theory ought to give a reliable picture of the surrounding airflow.

The first of the three tests used downwash data taken from a BE2E biplane with its wings at an angle of incidence of 6°. Measurements were made along an axis that was normal to the wing at its midpoint. For distances away from the wing of greater than one and a half times the chord, it was found that the predictions based on a uniform loading agreed fairly well.71 Other results, however, using wind-channel data from a monoplane wing with an RAF 6 section at an incidence of 3°, showed a downwash that was much greater than predicted. The second test measured downwash along the longitudinal axis, that is, at a number of points toward, and beyond, where the tailplane is typi­cally located. Observed values of the downwash were progressively smaller than those predicted by the elliptical distribution of lift but larger than those to be expected on uniform distribution. The trend of the results was roughly right but not the numerical values. The third test concerned the flow at the wingtips, and theoretical calculations were compared with wind-channel measurements made on a model Bristol fighter. This time, only calculations based on the elliptical distribution were used. (A uniform distribution was ruled out because it failed to represent the fact that lift falls to zero at the tips.) The predictions agreed with the observations in showing that, as one moves along the span of the wing, downwash decreases toward the tips and turns into an upwash beyond the tips.72

Glauert admitted that he was perplexed by the mixed results of the first test but deemed the results for the flow round the wingtips “quite good.” The theory represented the flow “with reasonable accuracy,” especially given all the approximations involved.73 The results for downwash on the longitudinal axis obviously took Glauert into the area studied by Foppl in the very first published test of Prandtl’s theory. Glauert concluded that the theory “cannot be used in any simple manner to predict the angle of downwash behind the wings”—which is exactly what Foppl had tried to do. The operative words, though, are “in any simple manner.” Glauert pointed out that the inaccuracy probably arose because the vortex sheet behind the wing was unstable and so the theory must be made more complicated to allow for this effect. He then noted that Prandtl had offered some suggestions about how to describe the rolling up of the vortex sheet. These promised to bring calculation and obser­vation back into alignment.

The study of the downwash behind a wing structure had held the promise of giving “a direct method of testing the underlying assumptions of the the­ory,” but it proved to be a complicated phenomenon and generated a lengthy and ramified program of experimental and theoretical investigation.74 Glau – ert was clearly sensitive to the problematic character of the empirical data and the complex relation between theoretical calculation and experimental measurement. It is also clear that he did not treat the empirical difficulties confronting Prandtl’s theory as refutations of the theory. He saw them as challenges that called for its further development. In a quiet but determined way Glauert shouldered the burden of developing the theory mathematically, and he did so, for a while, almost single-handedly. Farren and Tizard said Glauert was a “bonny fighter” in argument and worthy of any opponent, but they remembered him as a man of “essential modesty and gentleness.”75 This characterization accords with the calm and nonpolemical character of every­thing he wrote. Not all of those who came to support Prandtl shared these character traits. One who did not was the redoubtable Major Low.