Locating Kutta

Although Kutta’s 1902 thesis appears to have been lost, the historian Ulf Hashagen has discovered the examiner’s report in the archives of the THM. Hashagen draws attention to the revealing way in which Kutta’s work was described.35 Finsterwalder, who acted as both adviser and examiner, said that Kutta’s thesis was solidly constructed, industrious, and skillful. The calcula­tions presented many difficulties, and these required a detailed knowledge of the theory of functions. But, he went on,

Erfreulich ist aber auch, dafi die Aufgabe noch ein besonderes praktisches In – teresse besitzt—wie denn die von Kutta ins Auge gefafite kunftige Lehrtatig – keit gerade auf die Anwendungen der Mathematik sich beziehen soll. Dabei zeigt die gegenwartige Arbeit, wie auch die fruheren, recht guten von ihm verfafiten Abhandlungen und Kuttas ganzer Studiengang, dafi er die “Anwen­dungen der Mathematik” in dem modernen Sinne kennt, welcher sich mit wirklich aktuellen Fragen der Physik und Mechanik beschaftigt, statt—wie dies fruher ublich war—nur eine physikalische Einkleidung rein mathemati – scher Untersuchung unter “Angewandter Mathematik” zu verstehen. (257)

It was also good that the task had a specific practical interest—so it will be just the thing for Kutta to use in the future teaching that he intends to do on the application of mathematics. Like the very good earlier papers that he has authored, and indeed like his whole course of study, this work shows that he knows what it is to “apply mathematics,” in the modern sense of those words.

This means getting involved with real questions in physics and mechanics

rather than—as used to be the case earlier—dressing up pure mathematical

investigations in physical clothing and calling it “applied mathematics.”

Some mathematicians had been less than genuine in their attempt to accom­modate, or be seen to accommodate, the demands for relevance coming from the engineers in the technische Hochschulen. Finsterwalder wanted a real en­gagement. For Finsterwalder, applied mathematics in the “modern sense” would truly embody the ideals of a scientifically oriented technology, and Kutta’s aerodynamic work was held up as an exemplary case of the study of mechanics pursued in this spirit.36

Finsterwalder’s description of the modern spirit sometimes took on a de – tectably marshal air, as did that of his respected Munich colleague August Foppl.37 The period around 1910 was one of diplomatic crises and inter­national tension. It was also a time of intense public interest in aviation. There was the triumphant, but troubled, development of Germany’s giant Zeppelin airships and Bleriot’s dramatic flight across the English Channel.38 The military significance of these events would not have been lost on the at­tentive Finsterwalder. In the 1909 article on the scientific basis of aeronautics, cited by Kutta, Finsterwalder had already spoken ominously of “the demands of the time” (“Forderungen der Zeit”) and the “honorable contest of nations” (“edlen Wettstreit der Nationen”; 32). These were euphemisms for war. Fin- sterwalder had also noted, pointedly, that in aeronautical matters, unlike nautical ones, all nations could participate equally (“alle Staaten gleichmafiig beteiligt sind”; 31). Given Germany’s well-publicized naval arms race with Britain, this comparison could not have been a casual one. It carried the sug­gestion that what was proving expensive and difficult for Germany in the maritime sphere might be achieved more easily in the sphere of aeronautics because all nations had the same starting point. No wonder the worldly Fin- sterwalder was now encouraging Kutta to take up aerodynamics once again and prepare an extended version of his old research for publication.

Kutta’s work has achieved a classic status, and he has been accorded epon­ymous honor. The law relating lift and circulation bears his name, and the condition of smooth flow at the trailing edge is frequently referred to as the Kutta condition.39 His identification of a wing with a geometrical arc can be seen as a precursor to what is called the theory of thin aerofoils.40 But Kutta’s mathematical techniques have not entered the textbook tradition. His work is never described in his own terms; it is always reworked by means of later techniques. Finsterwalder did, however, encourage Wilhelm Deimler, an assistant in Munich, to calculate and publish the precise pattern of stream­lines around Kutta’s arc-like wing.41 Apart from this, and one unpublished doctoral dissertation, there appears to have been little by way of follow up.42

Looking at Kutta in retrospect, we may also remark that the empirical support he claimed is open to question. The quality of the data he used was not good. Kutta was aware of some of these problems, though not all of them. One obvious problem was that Kutta’s theory was two-dimensional, whereas Lilienthal’s data were three-dimensional. Lilienthal’s experiments were also conducted in a natural wind and therefore depended on the calibration of the anemometer. Unfortunately, Lilienthal was depending on inaccurate data from other experimenters to enable him to read the wind speed. Finally, when Kutta applied his own theory to a flat plate, he assumed a smooth flow in which the air stayed close to the surface, but the flat-plate data from Lang­ley and others did not satisfy this condition.

These shortcomings have been identified in Early Developments of Modern Aerodynamics by Ackroyd, Axcell, and Ruban. The book provides a transla­tion of Kutta’s papers of 1902 and 1910 and a valuable commentary from the standpoint of today’s aerodynamics.43 The commentary contains a brief, and negative, evaluation of Kutta’s discussion of the rounding off of the leading edge. Kutta is said to have embarked on this project “rather fruitlessly, as subsequent events were to prove” (185). Looking back with the benefit of hindsight this is fair comment. For the purpose of locating Kutta historically, however, it is important not to pass over the clue that this expenditure of ef­fort can give us, however misguided it now seems. In this part of his paper Kutta was engaging with a question that confronted those who were design­ing and building wings. What sort of leading edge should they give the wing? Fruitful or fruitless, Kutta’s attempt to grapple with this question provides evidence of the engineering orientation that Finsterwalder wanted. Although today’s reader may be tempted to hurry past these sections of Kutta’s paper, for Finsterwalder they were evidence of the intimate relation between Kutta’s mathematics and technology.44

The point deserves emphasis. If we are to understand Kutta historically, we must keep in mind the following characteristics of his work: (a) its focus on a specific, technical artifact, namely, the wing of Lilienthal’s glider; (b) his concern with issues of optimization and trade-off, for example, with regard to the amount of rounding off of the leading edge; (c) his willingness to use highly artificial theoretical tools such as perfect fluid theory; (d) his aware­ness of the limitations of these conceptual tools but a willingness to postpone asking and answering certain questions, for example, about the origin of the circulation; and (e) a determination to bring the theory, however unrealis­tic, into direct contact with data at the numerical level. This combination of mathematical methods with an engineering orientation placed Kutta’s work in the category that Finsterwalder called “modern” applied mathemat­ics. Also, when dealing with turbulence and other complications in the flow, Kutta, in his 1910 paper, used a version of what von Mises called the hydraulic hypothesis. Kutta went on:

Dennoch halte ich es fur moglich, dafi diese komplizierten Erscheinungen sich uber das hier beschriebene Stromungsbild nur superponieren, und die durchschnittlichen Druck-und Geschwindigkeitsverteilung—besonders die erstere—der geschilderten nahe steht. (51)

I therefore think it is possible that these complicated phenomena are merely superimposed on the flow formations described here, and the average pres­sure and speed distributions, especially the former, are close to those that have been presented.

Pioneering work also raises questions about its origins. Consider the break­through Kutta made in his 1902 Habilitationschrift. His appeal to the circula­tion theory of lift was made some years before the publication of Lanchester’s work. How then did Kutta come to make the link between circulation and lift? From where did he get his ideas? Of course, Finsterwalder may have pointed out the role of circulation when he suggested the research topic to Kutta, but this conjecture only postpones the problem. How did Finsterwalder get the idea?

The proper response is to see that there is something wrong with the ques­tion. Neither Kutta nor Finsterwalder needed to think up the idea. It was com­mon knowledge that a force can be generated by combining a uniform, recti­linear flow with a circulating flow. This had long been known in Cambridge and would have been equally well known in Munich. It was not the availabil­ity of the idea that constituted a problem; it was the willingness to use it. The thing that needs explaining is not how the idea was generated but why some people saw it as useful while others saw only trouble and futility. It is the mo­bilization of action that constitutes the real puzzle, not the origin of ideas.

Kutta clearly felt free to use the simple model of circulation in a perfect, irrotational fluid. He did not feel compelled to stop in his tracks because he could not explain how the circulation might arise. He knew that in using the circulation model he had to make physically false assumptions about viscos­ity. He might have been, but was not, deterred from going ahead on this basis. He might have concluded that he should have been devoting his efforts exclu­sively to a theory of viscous fluids, but he did not. This is the point. Kutta was prepared to go ahead, while others, such as G. I. Taylor in Cambridge, saw no point in taking this route. This was no personal idiosyncrasy on Kutta’s part, any more than the opposite response was a personal idiosyncrasy on Taylor’s part. Kutta was simply going with the flow or, at least, with the local flow. He was in a context where this course of action made sense and, if attended by a measure of success, would produce rewards rather than puzzlement.

Recall that August Foppl spoke of the necessity for the engineer to get answers, even if it meant using ideas that could not be justified within the broader framework of established, physical knowledge. One such scientifi­cally unjustifiable idea was that of the perfect, inviscid fluid. Here we have an explanation of the prima facie oddity that it was engineers who displayed the greater tolerance of ideal fluids, while the physicists proved intolerant of them. The engineer’s commitment to practicality does not mean that fictional concepts have to be shunned. Rather, it is the physicist’s commitment to truth that makes such fictions so unpalatable. In this respect, the engineer’s sense of necessity, and the shared awareness of this within the profession and its supporting institutions, generated a certain freedom of action. It meant that a pragmatic move or an expedient step did not meet with immediate criticism, provided its engineering utility could be demonstrated. On the contrary, if such a move contributed to the engineer’s project, it would be met with en­couragement. Encouragement was precisely what Finsterwalder gave to Kutta. Kutta’s willingness to venture where others would not venture becomes intel­ligible when set in the institutional context of technische Mechanik.