The Turbulence Problem pp 17-26 | Cite as

# The Turbulence Problem in the 1920s

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## Abstract

In the early 1920s the turbulence problem was perceived as the quest for a theory concerning the onset of turbulence. Its solution was expected along the Orr-Sommerfeld approach. It reached centre stage as a research subject for applied mechanics and mathematics. By the mid 1920s, however, Prandtl regarded fully developed turbulence as the greater turbulence problem for which he suggested the “mixing length” concept. A test case was the turbulent friction along a wall. Empirical data suggested at first that the mean velocity profile with growing distance from the wall obeys a power law; by 1929 it became clear that for high Reynolds numbers it was rather a logarithmic law. The derivation of the “wall law” became subject of a fierce rivalry between Prandtl and Kármán.

In the early 20th century “the true content of the problem of turbulence,” to resume Sommerfeld’s view from his Rome paper in 1908, had been perceived as the quest to determine the limit of stability of laminar flow. Thus it was expected to bridge the gap between theoretical hydrodynamics and practical hydraulics. The advent of aeronautics shifted the focus on aerodynamics—a research field whose lack of theoretical underpinning became exposed by wind tunnel investigations shortly before and during WW I. After the war, fluid mechanics became the wider umbrella for investigations of flow problems disregarding their occurrence in water, air or other fluids. By the same token, turbulence became regarded as a universal phenomenon beyond the former association with hydrodynamics, hydraulics and aerodynamics.

## 2.1 The Turbulence Problem in *ZAMM*

The main difference between the turbulence problem as perceived before and after WW I was that it became declared as a challenge for a new scientific community. Before the war, the annual Naturforscher meetings provided the only opportunity for applied mathematicians, physicists, and engineers in German speaking countries to express common concerns. When they met again for the first time after the war in 1920 in Bad Nauheim, Prandtl and others who perceived their experience from the war as a lesson for the future, expressed the need for a new umbrella for their research field. The result of their collaborative effort was the establishment of a new professional society, the Gesellschaft für angewandte Mathematik und Mechanik (GAMM), with Prandtl as president (Gericke 1972, pp. 5–10; Tobies 1982). Their mouthpiece was the *Zeitschrift für angewandte Mathematik und Mechanik (ZAMM)* founded shortly after the war and edited by Richard von Mises.

Already in the editorial to the first issue of *ZAMM* turbulence was declared as one of the themes within the scope of the journal, particularly problems about the circumstances responsible for the onset of turbulence (von Mises 1921, p. 12) . For more details, Mises hinted at Fritz Noether’s forthcoming paper on “The turbulence problem” assigned to the rubric of comprehensive review reports. “By the turbulence problem, generally speaking, we understand the question why Poiseuille flow, although it is always possible, is realized only within a restricted area and makes room for hydraulic flow.” Thus Noether defined the turbulence problem once more as the key to bridge the gap between hydrodynamics and hydraulics. He even confined “the use of the term ‘turbulent’ only for a certain area of transition between laminar and hydraulic flow.”^{1} Noether’s review portrayed the efforts to solve the turbulence problem by the method of small oscillations (i.e. what later was called the Orr-Sommerfeld approach) as a sequence of failures. The “energy method” introduced by Reynolds and Lorentz also appeared little promising. By and large, theory was in a dead end because it failed to predict a limit beyond which laminar flow would become unstable (Eckert 2010).

The same message was echoed from a different perspective in another *ZAMM*-paper on “Experimental investigations on the turbulence problem.” Its author was Ludwig Schiller, an experimental physicist from Leipzig University. “Up to now the numerous theoretical investigations about the ‘turbulence problem’ have not yet led to a satisfying result”, Schiller introduced his paper.^{2} He had just spent a longer research sojourn with Prandtl in Göttingen where he performed exhaustive experiments on pipe flow. The results became the subject of his habilitation thesis about “Investigations on laminar and turbulent flow in circular pipes” which promoted him in 1921 as Privatdozent for “physics and aeronautics” and in 1926 as extraordinary professor for “mechanics and thermodynamics” at the physics institute of Leipzig University.^{3} The allocations of Schiller’s specialty foreshadow the ambivalence of academic physics with regard to applied mechanics—an ambivalence that would also affect future research of physicists on turbulence. For the time being, Schiller’s *ZAMM*-paper provided evidence that the turbulence problem was not merely an issue for theorists. Based on the results of his habilitation he concluded that the critical Reynolds number for the transition from laminar to turbulent pipe flow could be as low as 200 and as large as 25500. For “technically smooth” pipes Schiller discerned the Reynolds number 1160 as a kind of true critical Reynolds number. Below this limit any vortices present in the inflow would always disappear within a sufficient calming zone. To each Reynolds number above 1160 “a certain disturbance is necessary in order to elicit turbulence. The larger the Reynolds number the lower the disturbance that is required to this.”^{4}

Schiller presented the results of his habilitation thesis also in September 1921 in Jena at a common conference of the Deutsche Mathematiker-Vereinigung, the Deutsche Physikalische Gesellschaft und the Deutsche Gesellschaft für Technische Physik where Prandtl and the likes of him invited “scientific engineers” to showcase their research on applied subjects in order to foster the foundation of what became the GAMM (Eckert 2017, Sect. 5.5). The turbulence problem thus became instrumental for the formation of a new scientific community. Prandtl added his own “Remarks on the genesis of turbulence” to the debate. They were published both in the new *ZAMM* (Prandtl 1921) and in the *Physikalische Zeitschrift* (Prandtl 1922). Prandtl’s remarks added to the turbulence problem a surprising aspect because “we found an instability of small oscillations contrary to the dogma.” What Prandtl called “the dogma” was the conclusion of almost all previous studies that the Orr-Sommerfeld approach failed to yield a critical Reynolds number, i.e. “the dogma” predicted stability for all Reynolds numbers (Eckert 2010). Prandtl and his doctoral student, Oskar Tietjens, arrived at the opposite result. “We could not believe this result at first and repeated the computation independently from another in three different ways. But there was always the same sign which meant instability.”^{5}

Both “the dogma” and Prandtl’s opposite conclusion, of course, were not compatible with the empirical evidence observed in real flows. Prandtl’s presentation was followed by a vivid debate with Sommerfeld, Kármán and others (Prandtl 1922, pp. 5–6) . Previously “the turbulence problem” was that all theories failed to yield a transition to turbulent flow up to the highest Reynolds numbers; now it was the opposite, that there should not even exist a laminar flow because the slightest disturbance would make it unstable even at the lowest Reynolds numbers. The presentations on “the turbulence problem” arose “great interest,” a reviewer highlighted the debate in *ZAMM*, and the Jena meeting as a whole “completely fulfilled the expectation that here, for the first time, within the annual meetings of mathematicians and physicists applied mathematics and mechanics showed to advantage to a larger extent and with one accord.”^{6}

## 2.2 A New International Forum for Applied Mechanics

In August 1922, *ZAMM* announced for the following month an international “hydro-aerodynamic conference” in Innsbruck (Austria).^{7} It was largely the result of a private initiative, conceived and carried through by Theodore von Kármán. After his return from the war, Kármán was eager to establish at the Technische Hochschule Aachen a budding center for modern fluid mechanics along the role model of Prandtl’s Göttingen institute. He shared with Prandtl also the ambition to foster applied mechanics as a rewarding challenge for scientific engineers. As he wrote in April 1922 to the mathematician Tullio Levi-Civita in Italy, “unfortunately the personal intercourse among those who work in this area is rather sparse.” Thus he attempted to extend the initiative that had just resulted in the formation of *ZAMM* and GAMM beyond the national borders of Germany. He suggested to convene “a very unofficial meeting” at Innsbruck as a location which could be regarded so shortly after the war as “neutral soil”. If “such a casual meeting” would be successful, more official international conferences could follow. He asked Levi-Civita to invite the colleagues from the Romance-language- and English-speaking countries, while he would care for those in Germany, the Netherlands, Austria, Switzerland, Russia, Czechoslovakia, and Scandinavia. He emphasized that he would extend the invitation to England and France, “but I cannot judge whether in these countries the friendly attitude is far enough advanced so that an invitation would not be rejected.”^{8}

Overcoming national resentments few years after the war proved difficult, but the success of the Innsbruck meeting added to the momentum to form a truly international community. Kármán and Johannes Martinus Burgers, the director of the Laboratory for Aero- and Hydrodynamics at the technical university in Delft, felt encouraged to plan a sequel mechanics congress with a more official character.^{9} This congress was held in September 1924 in Delft. Altogether there were 214 participants from 21 countries—a remarkable demonstration of international scientific cooperation at a time when science was still devided into hostile camps between the former Central Powers and the Entente (Kevles 1971). *ZAMM* reported about this “first international congress for applied mechanics” enthusiastically and announced the resolution of the congress committee to organize the next such event in 1926 in Zurich and then every four years in another country.^{10}

*R*for which it is possible.”

^{11}Two years later the turbulence problem became the subject of his doctoral dissertation

^{12}:

Thus the turbulence problem was no longer confined to the onset of turbulence only. Schiller shared this broader view when he argued that “almost all fluid motions that appear in practice or motions of solid bodies in air or in fluids belong to the ‘turbulence problem’. This wider definition contrasts with the relatively narrow meaning of the ‘turbulence problem’ in the mathematical-physical literature”, which should rather be called “the stability problem of hydrodynamics”. But he observed that among theorists also a wider notion was adopted. “Nowadays one has extended in the mathematically oriented literature the notion of the ‘turbulence problem’ insofar as one has added to the former problem that of fully developed turbulent motion.”For our purpose it is sufficient to provide a very rough sketch about the present state of the turbulence problem. The investigations so far may be categorized in two groups: the investigations of the first group deal with the stability analysis of some laminar motion, the other with turbulent motion itself.

^{13}

## 2.3 The “Great Problem of Developed Turbulence”

The tendency to extend the notion of the “turbulence problem” became most apparent when Prandtl reported by the same time in *ZAMM* about a new approach with which he attempted “to compute the distribution of the main flow in turbulent motion under various conditions hydrodynamically.”^{14} This approach became the subject of his presentation at the forthcoming international mechanics congress held in 1926 in Zurich. There Prandtl focused on “what I should like to call the ‘great problem of developed turbulence’, a deeper understanding and a quantitative computation of processes by which existing eddies despite their damping by friction create over and over new ones”. But he expected that this problem “will perhaps not so soon be solved.”^{15}

Prandtl’s approach became known as the mixing length concept. It was based on the introduction of a characterisic length which played a similar role for turbulence as the free path length in the kinetic theory of gases. Prandtl interpreted this length as the path that an eddy moves through the turbulent flow until it looses its identity by mixing with neighbouring parts of the fluid. “According to this meaning we call it mixing length and designate it by *l*.” He assumed that *l* is a mean length of motion in direction *y* perpendicular to the main flow in *x* direction with velocity \(\bar{u}\), and that in first approximation the velocity of an eddy in this transversal motion has a velocity that equals \(l \frac{\partial {\bar{u}}}{\partial {y}} \). Using this expression in Boussinesq’s eddy viscosity— by dimension the product of a velocity with a length—yielded the turbulent shear stress in terms of *l* (Prandtl 1927, pp. 62–64) .

In contrast to earlier attempts to account for turbulence in terms of Boussinesq’s eddy viscosity (see, e.g., Hahn et al. 1904, pp. 62–64), *l* could be adjusted to the boundary conditions of a specific turbulent flow and in general be assumed to be a function of spatial variables such as the distance from the wall in pipe or channel flow. In these cases, however, the approach met with problems (see Sect. 2.5). Therefore, Prandtl used at first an example of “free turbulence” without walls, such as the broadening of a turbulent jet ejected in an ambient fluid at rest. In this case he assumed that the mixing length increases proportional to the width of the jet in each cross-section behind the nozzle. He assigned the computation to his student Walter Tollmien who arrived at results in excellent agreement with experimental measurements. Thus the mixing length offered at least in one case a viable route to solve the turbulence problem “hydrodynamically”, although this did not mean that the theory was based on first principles only. The crucial assumption about the mixing length could only be made plausible retrospectively by comparison with experiments.^{16}

Prandtl’s mixing length concept was the result of that part of his “Working program for a theory of turbulence” which was dedicated to fully developed turbulence (see Sect. 1.5). He had penned it ten years ago in the midst of WW I, but put on the agenda of his institute only “about five years ago”, as he mentioned in the introduction to his Zurich lecture. Furthermore, despite the goal of “a theory of turbulence,” the program involved experiments. At Zurich he presented photographs and even a film that showed the turbulent flow in a channel. His expectation that the problem “will perhaps not so soon be solved” was not the least due to these experiments: “our photographic and cimematographic records only show how hopelessly complex these motions are even in the case of smaller Reynolds numbers.”^{17}

## 2.4 Tollmien’s Solution of the “Stability Problem”

While Prandtl was aiming with the mixing length concept at the “great problem of developed turbulence,” he did not loose sight of the other part of the turbulence problem concerned with the onset of turbulence, which Schiller had named more appropriately the “stability problem”. Prandtl’s presentation at the Jena meeting in 1921 and the detailed elaboration by his doctoral student Oskar Tietjens in *ZAMM* (Tietjens 1925) did not seem to offer a loophole through which the deadlock of the Orr-Sommerfeld approach could be overcome. But Prandtl must have regarded Tollmien’s successful application of the mixing length concept to turbulent jet broadening as a recommendation to assign Tollmien now the stability problem as subject of his doctoral work.

^{18}The “disturbance parameter” was proportional to the wave number of the disturbance (Fig. 2.1 ). In other words, the instability which preceded the onset of turbulence depended both on Reynolds number and disturbance in a way that made it difficult to discern a clear-cut origin of turbulence.

^{19}:

With the latter remark Tollmien alluded to an ongoing rivalry between Prandtl and Kármán about the theory of the turbulent boundary layer which became known as the quest for a “universal law of turbulence”.Firstly, because one does not know about the disturbances that actually occur; then it is not clear how far the point of transition as defined by these authors coincides with our beginning of instability of the laminar flow. Burgers, and similarly Hansen, define the point of transition in this way: In the front part of the plate there is a laminar flow which develops according to Prandtl-Blasius; beyond the point of transition there is a pronounced turbulent part whose laws have been elucidated by Prandtl and von Kármán (1/7 power law etc).

## 2.5 The Quest for a Universal Law of Turbulence

The rivalry had begun in the early 1920s. “Dear Master, colleague, and former boss,” Kármán had introduced in 1921 a five-page letter to Prandtl in which he sketched “a kind of ‘turbulent boundary layer theory’.”^{20} He derived the velocity profile in turbulent wall-bounded flows, \(U \sim y^{1/7}\), where *U* is the mean velocity parallel to a flat plate and *y* the distance perpendicular to the surface of the plate. From the published version of the theory (von Kármán 1921) it becomes clear why he designated his considerations only as “a kind of’turbulent boundary layer theory’.” The 1/7th power law was not derived from first principles but based on the empirical law of turbulent pipe flow established by Blasius in 1913, whereupon the loss of pressure per unit length varies like \(U^{7/4}\) (see Sect. 1.4). The derivation started from Blasius empirical formula and employed dimensional analysis in order to extrapolate from pipe flow to the velocity profile on a flat plate.

Prandtl had arrived at the same result earlier without publishing it. He had mentioned it in informal talks, as Kármán recalled in his letter. Prandtl’s derivation of the 1/7th law appeared in print only five years later in the doctoral dissertation of Johann Nikuradse to whom Prandtl had assigned experimental investigations of turbulent flow in a water channel. Nikuradse’s measurements agreed with the 1/7th law (Nikuradse 1926), but this corroboration contributed little to the theory of the turbulent boundary layer; it merely confirmed Blasius’s empirical formula upon which the 1/7th law was based. “You ask for the theoretical derivation of Blasius’ law for pipe friction,” Prandtl once responded to the question of a colleague. “The one who will find it will thereby become a famous man!”^{21}

When Prandtl developed by the same time his mixing length approach he hoped that it would serve as a new starting point for the theory of the turbulent boundary. However, he failed to derive the 1/7th law from one or another plausible assumption for the mixing length. He ruled out a linear relation between mixing length and distance from the wall, like in the case of the broadening of a turbulent jet (see Sect. 2.3), because this would have resulted in a logarithmic law with a singularity at the wall. Prandtl therefore dismissed the most plausible linear relation for the mixing length. In a lecture delivered in October 1929 in Tokyo during a trip around the world he argued: “The approach *l* proportional *y* does not lead to the desired result because it would yield \(\bar{u}\) prop. \(\log y\).”^{22} By the same time, however, Nikuradse concluded from new measurements on pipe flow at very high Reynolds numbers that a logarithmic formula yielded indeed a better agreement with the data than a power law (Nikuradse 1930) .

In the meantime, Kármán responded to the quest for a theory of wall turbulence from a new angle. In 1928, his doctoral student Walter Fritsch published the results of experimental investigations about turbulent channel flow with different wall surfaces; they showed that the velocity profiles could be superimposed upon one another if the shear stress at the wall was the same, regardless of the roughness of the wall surface (Fritsch 1928) . From this observation, Kármán concluded that at some distance from a wall the velocity fluctuations are similar anywhere and anytime in fully developed turbulent flow. Therefore, he regarded the mixing length as a characteristic scale of the fluctuating velocities. “The only important constant thereby is the proportionality factor in the vicinity of the wall,” he announced his theory in a letter to Burgers.^{23} Instead of the 1/7th law, Kármán’s similarity approach yielded a logarithmic wall law (von Karman 1930, 1931). The singularity did not bother Kármán because his approach was not valid immediately at the wall. Nevertheless he regarded his theory as universal because the only constant that entered his approach was independent of the experimental conditions. It was soon named after him the “Kármán constant”.

Prandtl swiftly confirmed Kármán’s results. He published his analysis in 1932 as a corollary to the presentation of new data in the series of his laboratory communications, *Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttingen*. He duly acknowledged Kármán’s earlier publications from the year 1930, but he claimed that he had arrived at the same results “at a time when Kármán’s papers had not yet been known, so that once more, like ten years ago with the same problem, the thoughts in Aachen and Göttingen followed parallel paths”.^{24} In Kármán’s recollection, however, Prandtl was “crestfallen” and “chagrined” that his former pupil had once more succeeded “in exercising his well-known talent for skimming the cream off the milk.” The quest for a “universal law of turbulence” had turned into a “first-class rivalry” (von Kármán 1967, pp. 135–138).

According to the dates of Kármán’s and Prandtl’s publications it is obvious that Kármán was the winner in this race. As we learn from their contemporary correspondence, however, the rivalry was not just about priority. The bone of contention became a report in the journal *Werft, Reederei, Hafen* about a conference at the Hamburgische Schiffbau-Versuchsanstalt where Prandtl’s results were presented as the authoritative state of the art. In a normal priority dispute it would have been sufficient for Kármán to point to his publications from the year 1930 in order to correct this erroneous view. But “who reads G. N. (Göttinger Nachrichten) and Stockholm Kongress,” Kármán complained in a letter to Prandtl; he insisted that Prandtl acknowledged his contribution in the *Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttingen* as the “standard for practitioners.”^{25} Kármán wished that his achievement was regarded as a milestone to practical engineering rather than pure science! Both Prandtl and Kármán regarded the “practitioners” as their true peers, i.e. the engineers of hydraulic and aeronautical laboratories. In other words: They perceived their approaches towards the turbulence problem, such as the derivation of the “universal law of turbulence” for wall-bounded flow, first and foremost as contributions to engineering rather than as solutions of the last riddle of classical mechanics.

## Footnotes

- 1.
Noether (1921, p. 126). Translation ME.

- 2.
Schiller (1921, p. 436). Translation ME.

- 3.
Personal File L. Schiller, PA 0254, University Archive Leipzig.

- 4.
Schiller (1921, p. 443) . Translation ME.

- 5.
Prandtl (1921, p. 434) . Translation ME.

- 6.
*ZAMM*1, 1921, pp. 419–420. Translation ME. - 7.
*ZAMM*2, 1922, p. 322. - 8.
Kármán to Levi-Civita, 12 April 1922. TKC 18.8. Translation ME.

- 9.
Burgers to Kármán, 15 May 1923. TKC 4.21.

- 10.
*ZAMM*3, 1924, pp. 272–276. For more details about the birth of the international mechanics congresses see Battimelli (1988) and Eckert (2006, Sect. 4.3). - 11.
Heisenberg (1922, p. 139) . Translation ME.

- 12.
Heisenberg (1924, p. 577) . Translation ME.

- 13.
Schiller (1925, pp. 566–567) . Translation ME.

- 14.
Prandtl (1925, p. 137) . Translation ME.

- 15.
Prandtl (1927, p. 62) . Translation ME.

- 16.
The theory of jet broadening was published in

*ZAMM*(Tollmien 1926) . For more details on the genesis of the mixing length approach see Eckert (2006, Sect. 5.3) and Bodenschatz and Eckert (2011, pp. 54–56). - 17.
Prandtl (1927, p. 62) . Translation ME.

- 18.
Tollmien (1929, p. 42) . Translation ME.

- 19.
Tollmien (1929, p. 43) . Translation ME.

- 20.
Kármán to Prandtl, 12 February 1921. GOAR 3684. Translation ME.

- 21.
Prandtl to Birnbaum, 7 June 1923. MPGA, Abt. III, Rep. 61, Nr. 137. Translation ME.

- 22.
Prandtl (1930, p. 9) . Translation ME.

- 23.
Kármán to Burgers, 12 December 1929. TKC 4.22. Translation ME.

- 24.
Prandtl (1932, p. 21). Translation ME.

- 25.
Kármán to Prandtl, 26 September 1932. AMPG, Abt. III, Rep. 61, Nr. 793. Translation ME. For more detail on this rivalry see Eckert (2017, Sect. 6.8).