# Prandtl, Ludwig

# PRANDTL, LUDWIG

(*b.*. Freising, Germany, 4 February 1875; *d*. Göttingen, Germany, 15 August 1953)

Prandtl was the founder of boundary layer theory and the originator of the German school of aerodynamics. His own work and that of his many students over half a century made Göttingen University the source of most of the elements of modern fluid mechanics. The notion of the boundary layer, the role of bound vortices in airfoil theory, the foundations of supersonic aerodynamics, explanations of drag coefficients and friction factors for a host of situations, the concept and applications of the turbulent mixing length, and the theory of wakes were results of his work.

Prandtl was the only child of Alexander Prandtl, an engineering professor at the agricultural college at Weihenstephan. Owing to the protracted illness of his mother, the former Magdalene Ostermann, he was particularly close to his father, who led him early in life to an interest in natural phenomena.

In 1894 Prandtl entered the Technische Hochschule at Munich to study engineering. He graduated in 1898 and two years later completed his doctorate under the famous mechanics professor, August Föppl, who had been one of his undergraduate teachers. His thesis dealt with the lateral instability of beams in bending. Although the mechanics of solids was to become a secondary interest, it was an area in which Prandtl continued to contribute significantly. He discovered the well-known soap film analogy for the Saint Venant torsion problem in 1904 and did fundamental work in plastic deformation during the early 1920’s.

Prandtl’s interest in fluid flow began immediately after his graduation from Munich. He went to work in the Maschinenfabrik Augsburg-Nürnberg, where he was asked to improve a suction device for the removal of shavings. In the process of greatly improving the device, he came to recognize some basic weaknesses in the current understanding of fluid mechanics.

One weakness was the inability of existing fluid mechanics to explain why the moving fluid in a pipe would separate from the wall in a sharply divergent section instead of expanding to fill the pipe. In the course of the next three years Prandtl attacked this problem in a way that was to be characteristic of his later work. After accepting a professorship at the Hannover Technische Hochschule in 1901, he proceeded to ask what the missing element in the existing analyses of pipe flows might have been. By 1904 he had developed his celebrated paper on the flow of fluids with small viscosity. In it he showed that no matter how small the viscosity was, the fluid had to be stationary on the walls of the pipe. Thus the classical theory of inviscid fluids could never be employed without taking cognizance of the way in which a thin viscous region near the wall shaped the flow. An understanding of this region, the boundary layer, was to facilitate the subsequent explanation of lift and drag on airfoils, and aspects of almost all other aerodynamic behavior. Prandtl’s insight thus led him to the large problem that lay behind a small one, and he set in motion a program of theoretical and experimental research that is still being worked out today.

By this time a new wind had begun to blow through the German system of higher education. The mathematician Felix Klein, feeling that the gulf between mathematics and technology was too wide, had established several technical institutes at Göttingen University. On Föppl’s recommendation Klein gave Prandtl a chair at Göttingen and placed him in charge of the Institute for Technical Physics. Just after this move Prandtl’s presentation of the boundary layer paper at the Third International Congress of Mathematicians at Heidelberg won high praise from Klein, who was quick to understand what his new man had accomplished.

At Göttingen, Prandtl had very good graduate students and access to the resources and interest of industry. He also was encouraged to pursue more theoretical lines of research than he had been at Hannover. During his first years there he made lasting contributions to the theory of supersonic flow. He combined Riemann’s theories with Mach’s *Schlieren* flow visualization apparatus to obtain the first explanation of the behavior of supersonic nozzle efflux. Later he was responsible for the first mathematical description of supersonic flow around slender bodies. He also was instrumental in developing the first German wind tunnel, which was completed at Göttingen in 1909.

Meanwhile, in the autumn of 1906, Prandtl’s most notable student arrived in Göttingen. Theodore von Kármán sought Prandtl out to direct his research on the theory of nonelastic buckling. He was only six years Prandtrl’s junior; and their lives and pursuits touched many times between 1906 and their last meeting in 1945, when Kármán, a Hungarian Jew who had become an American citizen, returned to Germany with an army interrogation team.

In his autobiography Kármán gives a picture of Prandtl that clearly mingles affection with annoyance. Prandtl’s life was, he tells us, “particularly full of overtones of naïveté.” In 1909, for example, Prandtl decided he really ought to marry; but he didn’t know how know how to proceed. Finally he wrote to Mrs. Föppl, asking for the hand of one of her daughters. But which one? Prandtl had not specified. At a family conference the Föppls made the practical decision that he should marry their eldest daughter, Gertrude. He did and the marriage was apparently a happy one. Two daughters were born in 1914 and 1917.

Another facet of his naïveté lay in Prandtl’s inclination to absorb himself totally in what interested him. The details of toys and magic tricks fascinated him to the exclusion of his surroundings. His greatness as a teacher did not include greatness as a lecturer because “he could not make a statement without qualifying it” and consequently was tedious. Nevertheless Prandtl was personable, gracious, and unassuming. He was an accomplished pianist with a good musical sense. His importance as a researcher was matched by an extraordinary ability to work fruitfully with his individual students.

Prandtl’s aerodynamic work developed steadily from 1909 to the end of World War I. Between 1909 and 1912 he helped to establish test codes for fans. In 1914 he explained a puzzling, sudden drop of the drag coefficient for a sphere that occurred with increasing velocity. He did this by again recognizing the important phenomenon underlying a minor problem: the laminar-to-turbulent transition of the boundary layer on the body.

Prandtl’s most significant contribution during this period was his work on airfoil theory. The lift force on wings was fairly well understood, and he turned to the explanation of drag. The boundary layer gave rise directly to skin friction which was much too small to account for wing drag. In 1911 and 1912 Kármán, stimulated by the experimental work going on in Prandtl’s laboratory, did much to explain another component, profile drag. He described the so-called Kármán street of alternating vortices, parallel with an aerodynamic body, that must be pulled along behind it. Prandtl continued to work on a third contribution, induced drag. He had recognized that the presence of lift causes a trailing vortex to be induced in the shape of a long distorted horseshoe with its base at the airport where the flight began and its ends at the wing tips, which continually generate the vortex.

Prandtl’s attempts to analyze this source of drag subsequently fell under the veil of wartime secrecy, and his descriptions of the effect finally emerged in restricted Göttingen publications in 1918 and 1919. By 1920 the idea reached a larger public and wrought sharp changes in the wing design and streamlining of airplanes. It led in general to more cleanly shaped wings, to higher aspect ratios, to today’s swept-back wings, and to the use of streamlining fillets.

The elusive problem of describing turbulent flow yielded to Prandtl’s and Kármán’s competitive efforts in the mid-1920’s. Kármán, who then was teaching at Aachen, discussed the structure of turbulent flows in 1924 but failed to produce a strategy for analyzing it. In 1926 Prandtl provided the conceptual device needed to make an analysis. This was the “mixing length,” or average distance that a swirling fluid element would travel before it dissipated its motion. The idea resembled the notion of a mean free path in the kinetic theory of gases and was used in a somewhat similar way. Subsequent experiments by Prandtl and theoretical work by Kármán made it possible for Prandtl to present a summary paper on the subject in 1933 that is the basis for chapters on turbulence in today’s textbooks.

During the 1930’s and 1940’s Prandtl, now an elder statesman of fluid mechanics, continued to contribute to the basic literature and technology, and reaped honors for his accomplishments. Nevertheless, his interrogation by the U.S. Army in 1945 found that the mainstream of wartime technology had more or less bypassed Prandtl and Göttingen. Hitler had shown greater interest in the rocketry at Peenemunde and the laboratory at Brunswick. The team, arriving in Göttingen from the slave labor camp at Nordhausen, found Prandtl complaining peevishly about bomb damage to his roof and asking how the Americans planned to support his ongoing work.

While the picture is one of a technician who felt little obligation to anyone’s politics, it forms a consistent view of Prandtl, who, after all, made enormous contributions to his world as a technologist. He was then seventy-one and just turning his attention to meteorology, on which he published material as late as 1950.

## BIBLIOGRAPHY

I. Original Works. Poggendorff, V, 1002; VI, 2069-2070; VIIa, pt. 3, 620, lists some 80 articles and books written between 1901 and 1950. Among the most important are “Zur Torsion von prismatischen Stäben,” in *Physikaische Zeitschrift*, **4** (1903), 758-767; “Über die stationären Wellen in einem Gasstrahl,” *ibid.*, **5** (1904), 599-601; “Über Flüssigkeits-Bewegung bei sehr kleiner Riebung,” in *Verhandlungen der III Internationaler MathematikerKongress* (Leipzig, 1905); “Die Luftwiderstand von Kugeln,” in *Nachrichten von der Geseltschaft der Wissenschaften zu Göttingen* (1914), 177-189; “Tragfliigel Theoric, 1 und 2, Mitteilungen,” *ibid.*, (1918), 451-477, and (1919), 107-137; “Über die ausgebildete Turbulenz,” in *Proceedings of the Second International Congress for Applied Mechanics* (Zurich–Leipzig, 1927), 62; and *Führer durch die Strömungslehre*, 3rd ed. (Brunswick, 1949).

II. Secondary Literature. Poggendorff, VIIa, pt. 3, 619-620, provides many references to writings about Prandtl and his work. Hermann Schlichting, *Boundary Layer Theory*, 4th ed. (New York, 1960), which includes 61 references to Prandtl and far more than that number to his students, provides the best overview of his impact on viscous flow theory. H. W. Liepmann and A. Roshko, *Elements of Gasdynamics* (New York, 1957), gives a comparable picture of his role in compressible flow theory. Additional biographical material appears in Kármán’s autobiography, *The Wind and Beyond* … (New York, 1967).

John H. Lienhard

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