Ludwig Prandtl

How to use this course

By 1900 hydrodynamics had split in two. Mathematicians studied ideal fluids without friction and proved that a body moving steadily through such a fluid feels no drag at all. Engineers used empirical tables that no theory explained. Ludwig Prandtl joined the two halves with one idea: friction matters only in a thin layer next to a surface. From that idea came the modern theories of drag, lift, flow separation and turbulence, and the Göttingen school that trained a generation of aerodynamicists.

Prandtl's career also ran through the whole Third Reich. He never joined the Nazi Party, yet he served Hermann Göring's air ministry at a senior level and defended the regime's persecution of Jews in private letters. Lesson 11 sets out that record from the documentary sources.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The order is roughly chronological: similarity and measurement, the boundary layer (1904), its first exact solution (1908), wing theory (1918–19), compressible flow (1908 onward), turbulence (1925 onward), and Prandtl's contributions to solid mechanics.

Timeline

Freising and Munich, 1875–1900

Ludwig Prandtl was born on 4 February 1875 in Freising, a cathedral town north of Munich. His father, Alexander Prandtl, taught at the agricultural college of Weihenstephan and did research on milk production. Several of Ludwig's siblings died young, and his mother, Magdalena Ostermann, suffered a long mental illness after these losses. The boy spent most of his time with his father, who encouraged him to watch natural phenomena closely and to think about what he saw.

He attended the Domberg Gymnasium in Freising from 1885 and the Ludwig Gymnasium in Munich from 1888, finishing in July 1894. That autumn he entered the Technische Hochschule in Munich to study mechanical engineering, and he graduated in 1898.

Föppl and the buckling thesis

The decisive teacher was August Föppl, professor of engineering mechanics, whose textbooks presented strength of materials as applied mathematics. Prandtl became his assistant in October 1898. The Technische Hochschule could not yet award doctorates, so his thesis went to the University of Munich. It treated the lateral buckling of a beam bent in its stiff plane, a problem in elastic stability, under the title Kipp-Erscheinungen, ein Fall von instabilem elastischem Gleichgewicht. The oral examination took place on 29 January 1900, with the mathematician Ferdinand von Lindemann among the examiners. Föppl remained a lifelong influence; nine years later he also became Prandtl's father-in-law.

Nuremberg and Hannover, 1900–1904

On 1 January 1900 Prandtl joined the Maschinenbaugesellschaft Nürnberg, part of the group that became MAN, as an engineer. One assignment was a suction system for removing shavings from machine tools. He noticed that the airflow in a widening duct did not follow the walls as ideal-flow theory predicted. It broke away from them, and the expected pressure recovery did not happen. This observation of flow separation stayed with him.

In 1901 the Technische Hochschule Hannover appointed him professor of mechanics, with support from the applied mathematician Carl Runge. At 26 he was the youngest professor in Prussia. In Hannover he built a small water channel and began to visualize flow around bodies with suspended particles. He was trying to understand why real fluids separate from surfaces and form wakes, while the classical theory of Euler and Laplace predicted smooth flow that closes behind a body. In 1903 he also published a membrane analogy for the torsion of bars, which Lesson 10 treats. By 1904 he had a physical picture of separation and a rough scaling argument for it.

Göttingen and the first wind tunnel, 1904–1914

On 8 August 1904 Prandtl spoke at the Third International Congress of Mathematicians in Heidelberg. His paper, Über Flüssigkeitsbewegung bei sehr kleiner Reibung (On fluid motion with very small friction), filled only eight printed pages. It introduced the thin friction layer next to a wall, showed how an adverse pressure gradient makes the layer separate, and included photographs from his water channel. Few listeners grasped its importance. Felix Klein did. Klein was building applied mathematics and engineering science at Göttingen, and he recruited Prandtl to head the department of technical physics there from 1 September 1904. In 1905 Prandtl and Runge jointly took over a new institute of applied mathematics and mechanics; Prandtl later headed its mechanics side and turned down a call to Stuttgart in exchange for a full professorship.

Airships and the wind tunnel

Germany's interest in Zeppelin airships gave Prandtl the means to test his ideas. In 1907 a study society for motor airships commissioned him to set up a model testing station, and in 1908 he completed a closed-circuit wind tunnel in Göttingen, in which air returns around a loop to the test section. The design gave steady, controllable flow and became standard worldwide as the Göttingen type.

His first doctoral students now produced the classical results of the school. Heinrich Blasius solved the boundary layer on a flat plate in 1908 (Lesson 06). Theodor Meyer computed supersonic expansion around a corner in the same year (Lesson 08). Theodore von Kármán took his doctorate in 1908 with a thesis on buckling and stayed as an assistant; his analysis of the alternating vortex street behind a cylinder dates from 1911. On 11 September 1909 Prandtl married Gertrud Föppl, Föppl's daughter. They had two daughters, Hildegard (born 1914) and Johanna (born 1917); Johanna Vogel-Prandtl later wrote a biography of her father. He was elected to the Göttingen Academy of Sciences in 1914.

Similarity and measurement

For the general reader. A wind tunnel model is small, but an aircraft is large. Prandtl's laboratory relied on a principle that lets one predict the full-scale forces from the model: two geometrically similar flows behave the same way when one number, the Reynolds number, is the same. Prandtl also gave engineers a practical instrument, the Prandtl tube, which reads flow speed from a pressure difference.

The Reynolds number

The incompressible Navier–Stokes equations for velocity \(\mathbf u\), pressure \(p\), density \(\rho\) and kinematic viscosity \(\nu\) are

Choose a length \(L\) and speed \(U\), and set \(\mathbf u=U\mathbf u^*\), \(\mathbf x=L\mathbf x^*\), \(t=(L/U)t^*\), \(p=\rho U^2p^*\). Dropping stars,

The principle goes back to Stokes and Osborne Reynolds; the Göttingen laboratory turned it into routine practice. A dramatic example came from spheres. Measurements at Göttingen and in Gustave Eiffel's Paris laboratory disagreed until it became clear that \(C_D\) for a sphere drops sharply near \(\mathrm{Re}\approx3\times10^5\), where the boundary layer becomes turbulent and separates later. In 1914 Prandtl showed that a thin wire ring placed on the front of a sphere triggers the same drop at lower speed, a direct demonstration that the boundary layer controls the drag.

The Prandtl tube

A Pitot-static tube has a forward-facing hole at the stagnation point, where the pressure is the total pressure \(p_0\), and side holes that sense the static pressure \(p\). For steady, incompressible flow along a streamline, Bernoulli's equation gives

Prandtl's version placed the static holes at a distance along the stem where the disturbances from the nose and the stem cancel, so the instrument needs no calibration factor. German engineers still call it the Prandtl tube.

The boundary layer

For the general reader. Air has very little viscosity, so for most of a flow it behaves as if it had none. Yet at a solid surface the air must stick and come to rest. Prandtl saw that the change from rest to full speed happens in a very thin layer. Inside it friction dominates; outside it can be ignored. When the layer is slowed too much by rising pressure, it lifts off the surface and the flow separates, which is the origin of most drag and of the stall of a wing.

d'Alembert's paradox

In an inviscid, irrotational, incompressible flow, a body moving steadily feels zero drag. This is d'Alembert's paradox. Setting \(\nu=0\) in Navier–Stokes lowers the order of the equations and drops the no-slip condition \(\mathbf u=0\) at the wall. Prandtl recognised the limit \(\nu\to0\) as singular: the viscous term cannot be neglected everywhere, however small \(\nu\) is.

Scaling

Consider steady two-dimensional flow along a wall \(y=0\), with \(x\) along the wall. Let the layer have thickness \(\delta\ll L\). Inside it, \(u\sim U\), \(\partial/\partial x\sim1/L\), \(\partial/\partial y\sim1/\delta\). Continuity \(u_x+v_y=0\) forces \(v\sim U\delta/L\). Balance the inertial term against the dominant viscous term:

Keeping the leading terms in each equation and noting that the \(y\)-momentum equation reduces to \(\partial p/\partial y\approx0\) gives Prandtl's equations.

The equations are parabolic in \(x\): information travels downstream, so one can march from a starting profile. This made them far easier to solve than the full Navier–Stokes system and created the method of matched inner and outer regions that later became singular perturbation theory.

Separation

At the wall \(u=v=0\), so the momentum equation gives the wall compatibility condition

In an adverse pressure gradient (\(dp/dx>0\)) the profile curves upward at the wall while it must curve downward far out, so it has an inflection point. Fluid near the wall, already slowed by friction, is pushed back by the rising pressure. Separation occurs where the wall shear stress \(\tau_w=\mu\,\partial u/\partial y|_{0}\) vanishes; downstream there is reverse flow. On the rear of a cylinder or the upper surface of a wing at high angle of attack this produces a broad wake, high drag and loss of lift.

Blasius and the flat plate

For the general reader. The simplest boundary layer forms on a thin flat plate set edgewise in a uniform stream. In 1908 Prandtl's first doctoral student, Heinrich Blasius, showed that the velocity profiles at different distances from the leading edge are all the same curve, stretched. The problem reduces to one ordinary differential equation, whose solution still serves as the benchmark for every boundary-layer calculation.

The similarity reduction

With \(U_e=U\) constant, there is no pressure gradient. Introduce a stream function \(\psi\) with \(u=\psi_y\), \(v=-\psi_x\), and the similarity variable

Then \(u=Uf'(\eta)\) and \(v=\tfrac12\sqrt{\nu U/x}\,(\eta f'-f)\). Substituting into \(uu_x+vu_y=\nu u_{yy}\) and simplifying gives

Numerically one integrates from \(\eta=0\) with a guessed \(f''(0)\) and adjusts it (shooting) until \(f'\to1\). The equation has a scaling symmetry: if \(f\) solves it, so does \(\lambda f(\lambda\eta)\). So one can integrate once with \(f''(0)=1\), read off the limit \(f'(\infty)=c\), and rescale with \(\lambda=c^{-1/2}\), avoiding iteration. The hero figure on this page uses the converged value.

Engineering results

With \(\mathrm{Re}_x=Ux/\nu\):

The drag grows like \(U^{3/2}\) and \(L^{1/2}\), a result first verified in the Göttingen tunnel. It applies only while the layer is laminar, typically up to \(\mathrm{Re}_x\) of a few hundred thousand.

Lifting-line theory

For the general reader. A wing lifts because it makes the air circulate around it. On a wing of finite span, this circulation cannot simply end at the tips. It trails behind as a sheet of vortices, which pushes the air down behind the wing and tilts the lift backward. The backward component is induced drag, the price of making lift. Prandtl's theory of 1918–19 computes it and shows that an elliptical spread of lift along the span makes it as small as possible.

From two to three dimensions

For two-dimensional flow past an aerofoil with circulation \(\Gamma\), the lift per unit span is given by the Kutta–Joukowski theorem, \(L'=\rho V\Gamma\) (Wilhelm Kutta 1902, Nikolai Joukowski 1906). Thin-aerofoil theory gives \(c_l=2\pi(\alpha-\alpha_{L0})\). Frederick Lanchester had described trailing tip vortices qualitatively in 1907. Prandtl, with Albert Betz and Max Munk, made the picture quantitative.

Replace the wing of span \(b\) by a bound vortex along \(-b/2\le y\le b/2\) whose strength \(\Gamma(y)\) varies along the span. By Helmholtz's vortex theorems, each change \(d\Gamma\) sheds a trailing vortex filament of strength \(-d\Gamma\) running downstream. By the Biot–Savart law, these filaments induce a downward velocity (downwash) at the lifting line

where P denotes the principal value. The local section sees an effective angle of attack \(\alpha_{\text{eff}}=\alpha-w/V\).

Proof sketch. Substitute \(y=\tfrac b2\cos\vartheta\) and expand \(\Gamma=2bV\sum_n A_n\sin n\vartheta\). Glauert's integral

gives \(w(\vartheta_0)=V\sum_n nA_n\sin n\vartheta_0/\sin\vartheta_0\). Then \(C_L=\pi AA_1\) and \(C_{D_i}=\pi A\sum_n nA_n^2=\frac{C_L^2}{\pi A}\big(1+\sum_{n\ge2}n A_n^2/A_1^2\big)\). The extra sum vanishes exactly when \(A_n=0\) for \(n\ge2\), the elliptic case. In 1933 Prandtl asked a different question, minimizing induced drag for a given lift and a given bending moment at the root, and found a bell-shaped loading with a longer span.

Compressible flow

For the general reader. At low speed air behaves as if it were incompressible. Near and above the speed of sound it does not: pressure waves pile up into shock waves, and supersonic flow can turn sharp corners by fanning out. Prandtl studied these effects from 1905 onward, first in steam nozzles and then for aircraft, and his group produced several of the basic formulas still in every textbook.

Subsonic flow: the Prandtl–Glauert rule

For a thin body in a stream of Mach number \(M_\infty<1\), write the velocity as \((V_\infty+\phi_x,\phi_y)\) with \(\phi\) small. The linearized potential equation is

With \(\beta=\sqrt{1-M_\infty^2}\) and the stretched coordinate \(\eta=\beta y\), this becomes Laplace's equation. Matching the boundary condition on the body leads to

Prandtl used the rule in Göttingen before Hermann Glauert published a derivation in 1928, hence the double name. It fails as \(M_\infty\to1\), where the linearization breaks down.

Supersonic expansion: Prandtl–Meyer

In 1908 Prandtl's student Theodor Meyer solved the flow of a supersonic stream around a convex corner. The flow turns through a centred fan of Mach waves with no losses, and the turning angle is a function of Mach number alone.

The derivation starts from the relation \(d\theta=\sqrt{M^2-1}\,dV/V\) along a Mach wave and integrates it using the energy equation. In 1929 Prandtl and Adolf Busemann turned the method of characteristics into a graphical procedure for designing supersonic nozzles.

Normal shocks

In the years after 1904 Prandtl photographed shock patterns in steam jets from Laval nozzles and studied the shock as a thin dissipative layer. For a normal shock in an ideal gas, with \(a^*\) the speed of sound at the state where the flow speed equals the sound speed, continuity, momentum and energy combine into the Prandtl relation

It shows at once that a normal shock takes a supersonic flow (\(u_1>a^*\)) to a subsonic one (\(u_2<a^*\)).

Turbulence, heat and stability

For the general reader. Most flows in engineering are turbulent, full of eddies that mix momentum and heat far faster than molecular friction does. Prandtl proposed in 1925 that an eddy carries momentum over a typical distance, the mixing length, before blending into its surroundings. This simple picture predicts the logarithmic velocity profile found near every wall. His school also explained how a smooth laminar boundary layer becomes unstable in the first place.

Mixing length

Split velocity into mean and fluctuation, \(u=\bar u+u'\). Averaging the Navier–Stokes equations adds the Reynolds stress \(-\rho\overline{u'v'}\) to the viscous stress. Prandtl modelled it by analogy with the kinetic theory of gases:

where the mixing length \(\ell\) is the distance a fluid parcel travels before losing its identity. Near a wall the only available length is the distance itself, so \(\ell=\kappa y\).

Derivation. With \(\tau_t=\tau_w\) and \(\ell=\kappa y\), \(\rho\kappa^2y^2(d\bar u/dy)^2=\tau_w\), so \(d\bar u/dy=u_\tau/(\kappa y)\). Integrating gives the logarithm; the constant \(B\) depends on the thin viscous sublayer and must come from experiment. Kármán reached the same law in 1930 by a similarity hypothesis. Johann Nikuradse's pipe-flow measurements in Prandtl's institute (1932–33), over smooth and sand-roughened pipes, fixed the constants and led to Prandtl's friction law for smooth pipes, \(1/\sqrt f=2.0\log_{10}(\mathrm{Re}\sqrt f)-0.8\).

The Prandtl number

In 1910 Prandtl extended Reynolds's analogy between heat transfer and wall friction. The ratio that governs it is

momentum diffusivity over thermal diffusivity. For air \(\mathrm{Pr}\approx0.7\); for water at room temperature about 7; for liquid metals far below 1. For laminar flow over a plate with \(\mathrm{Pr}\) not too small, the thermal layer is thinner than the velocity layer by roughly \(\mathrm{Pr}^{-1/3}\).

Transition

Rayleigh had shown that inviscid parallel flows without an inflection point are stable. Prandtl suspected that viscosity itself could destabilize a boundary layer. Walter Tollmien (1929) and Hermann Schlichting (1933) solved the Orr–Sommerfeld equation for boundary-layer profiles and found unstable travelling waves, now called Tollmien–Schlichting waves, above a critical Reynolds number, \(\mathrm{Re}_{\delta^*}\approx520\) for the Blasius profile. The theory was doubted until Galen Schubauer and Harold Skramstad observed the waves in a very quiet wind tunnel, in work published in 1947.

Solids: torsion, plasticity, friction

For the general reader. Prandtl trained as a mechanical engineer, and some of his most durable ideas concern solids. He showed that the stress in a twisted bar can be read off the shape of a soap film stretched over a hole of the same cross-section. He computed how hard one must press a flat punch to make metal flow. And he built a simple model of a sliding atom that is still the starting point of the physics of friction.

Torsion and the membrane analogy (1903)

For a prismatic bar twisted at rate \(\theta'\) (radians per unit length), with shear modulus \(G\), Prandtl introduced a stress function \(\phi(x,y)\) on the cross-section \(\Omega\) with \(\tau_{xz}=\partial\phi/\partial y\), \(\tau_{yz}=-\partial\phi/\partial x\).

The analogy lets one see where stresses concentrate. At a re-entrant corner the film is steep, so the stress is high; this is why shafts have rounded keyways.

Plastic indentation (1920)

For a rigid, perfectly plastic material obeying the Tresca or von Mises criterion with yield stress in shear \(k\), Prandtl constructed a slip-line field under a flat rigid punch in plane strain.

The same number reappears in soil mechanics as the bearing capacity factor of a strip foundation on cohesive soil.

The Prandtl–Tomlinson model (1928)

A particle of mass \(m\) is pulled through a periodic potential of period \(a\) by a spring of stiffness \(k_s\) whose end moves at speed \(v\):

If the spring is stiff, the particle glides smoothly and friction vanishes as \(v\to0\). If the potential is strong, the particle sticks in a well and then slips abruptly, dissipating energy even at vanishing speed. This stick-slip motion gives a static friction force from atomic-scale physics, and the model is used to interpret atomic force microscope experiments.

Institutes, war and the Third Reich, 1914–1945

Building the Göttingen institutions

The First World War turned Prandtl's test station into a military facility. A larger wind tunnel, started in 1915 and finished in 1917, tested aerofoils and aircraft components for the army's air service, and much of the wing theory of Lesson 07 was worked out in that setting. In 1919 the station was renamed the Aerodynamische Versuchsanstalt (AVA). In 1922 Prandtl co-founded the Gesellschaft für Angewandte Mathematik und Mechanik and presided over it until 1945. In 1925 the Kaiser Wilhelm Society opened its Institute for Flow Research in Göttingen under his direction, with the AVA attached. These were the years of the mixing length, of Nikuradse's pipe experiments, and of the textbooks written with Oskar Tietjens. International honours followed: Foreign Membership of the Royal Society in 1928 and the Daniel Guggenheim Medal in 1930.

The Nazi period: the documentary record

Historians, chiefly Michael Eckert in his biography based on Prandtl's papers, have documented his conduct after 1933. The main points are these.

• No party membership. Prandtl never joined the NSDAP and described himself as unpolitical. A Nazi assessment called him politically harmless, a scientist in an ivory tower.
• Interventions for colleagues. After the civil-service law of April 1933 he protested against the dismissal of Jewish and politically unwelcome colleagues, without success. He opposed the "German Physics" movement and supported Werner Heisenberg against its attacks, reportedly through direct contact with Heinrich Himmler. In 1941–42 he helped Carl Ramsauer prepare the German Physical Society's petition arguing that the rejection of modern physics was damaging German science.
• Expansion under the air ministry. Rearmament brought his institutes money on a new scale through Göring's Reich Air Ministry. The regime used his international standing to promote German science; Eckert's biography describes him as a goodwill ambassador for Nazi Germany. He received the Harnack Medal of the Kaiser Wilhelm Society and an honorary Cambridge doctorate in 1936, and the Goethe Medal for Art and Science in 1940.
• Private statements. In a 1937 letter to a representative of the American NACA he praised Italian Fascism and German National Socialism as good beginnings of a new way of thinking. In letters of 1938 and 1939 to Geoffrey Ingram Taylor and his wife he wrote that Germany's struggle against the Jews had been "necessary for its self-preservation", and after the outbreak of war he placed the guilt for it on Britain.
• War research. After 1939 he reviewed his institute's projects for military relevance; basic work such as turbulence research was cut back and funds moved to war projects. In 1942 he became chairman of the Reich research directorate for aeronautics, under the authority of Göring as air minister and commander of the Luftwaffe.

The record therefore combines real interventions for individuals and for the integrity of physics with service to an aggressive air force and explicit approval of antisemitic persecution. Eckert treats Prandtl as a prime example of how science and politics became entangled under National Socialism. Post-war denazification raised no objections against him, and his surviving post-war letters show no remorse; he saw himself as a research organizer who had served his country.

Last years and legacy, 1945–1953

American troops reached Göttingen in April 1945, and the town later fell in the British zone of occupation. Allied technical teams examined the AVA, and Theodore von Kármán, Prandtl's former student, who had left Germany for Caltech in 1930, came as a scientific adviser to the US Army Air Forces and questioned his teacher. Prandtl retired as director of the Kaiser Wilhelm Institute in 1946 but kept working, now on meteorology and on a revised edition of his textbook Führer durch die Strömungslehre, first published in 1942. The Kaiser Wilhelm Society was refounded as the Max Planck Society in 1948, and his institute became the Max Planck Institute for Flow Research; since 2004 it has been the Max Planck Institute for Dynamics and Self-Organization.

In November 1950 a stroke left him partly paralysed on one side, and a second followed in August 1952. He died in Göttingen on 15 August 1953, aged 78. In 1953 he was also awarded the Order of Merit of the Federal Republic of Germany.

Legacy

• Fluid mechanics: the boundary layer is the organizing concept of viscous flow at high Reynolds number, from aircraft and ship design to meteorology. Its mathematical form, an inner layer matched to an outer solution, became the method of matched asymptotic expansions.
• Aeronautics: lifting-line theory, induced drag and the elliptic wing, compressibility corrections and supersonic nozzle design were standard tools of aircraft design through the twentieth century.
• Turbulence: the mixing length and the log law remain the basis of engineering turbulence models and wall functions.
• The school: his students included Heinrich Blasius, Theodore von Kármán, Theodor Meyer, Max Munk, Jakob Ackeret, Adolf Busemann, Johann Nikuradse, Walter Tollmien and Hermann Schlichting. Through Kármán at Aachen and Caltech, Ackeret at Zürich and Munk at the NACA, Göttingen methods spread worldwide.
• Honours: the German aerospace society awards the Ludwig Prandtl Ring; a lunar crater bears his name; the Prandtl number appears in every heat-transfer calculation. The institution he founded, the AVA, is the oldest predecessor of the German Aerospace Center (DLR).

Sources and further reading

Biography and history

1. J. J. O'Connor and E. F. Robertson, Ludwig Prandtl, MacTutor History of Mathematics.
2. Ludwig Prandtl, Wikipedia; German Wikipedia (honours and academy memberships).
3. M. Eckert, Ludwig Prandtl: A Life for Fluid Mechanics and Aeronautical Research, Springer, 2019 (German original 2017). The main source for the Nazi period.
4. Max Planck Institute for Dynamics and Self-Organization, Ludwig Prandtl (institute history, including his role under National Socialism).
5. M. Eckert, The Dawn of Fluid Dynamics: A Discipline between Science and Technology, Wiley-VCH, 2006.
6. J. Vogel-Prandtl, Ludwig Prandtl: A Personal Biography Drawn from Memories of His Daughter, Universitätsverlag Göttingen, 2014.
7. A. Busemann, Ludwig Prandtl, Biographical Memoirs of Fellows of the Royal Society 5 (1960).
8. T. von Kármán with L. Edson, The Wind and Beyond, Little, Brown, 1967.
9. O. Darrigol, Worlds of Flow, Oxford University Press, 2005.

Physics

1. L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Verhandlungen des III. Internationalen Mathematiker-Kongresses, Heidelberg 1904, Teubner, 1905.
2. H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys. 56 (1908).
3. L. Prandtl, Tragflügeltheorie I und II, Nachr. Ges. Wiss. Göttingen (1918, 1919).
4. L. Prandtl, Bericht über Untersuchungen zur ausgebildeten Turbulenz, ZAMM 5 (1925).
5. L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, ZAMM 8 (1928).
6. J. D. Anderson, Ludwig Prandtl's boundary layer, Physics Today 58(12) (2005).
7. H. Schlichting and K. Gersten, Boundary-Layer Theory, 9th ed., Springer, 2017.
8. J. D. Anderson, Fundamentals of Aerodynamics, McGraw-Hill (lifting-line theory, Prandtl–Glauert rule, Prandtl–Meyer flow).
9. L. Prandtl, Essentials of Fluid Dynamics (translation of Führer durch die Strömungslehre), Blackie, 1952.
10. V. L. Popov and J. A. T. Gray, Prandtl–Tomlinson model: history and applications in friction, plasticity, and nanotechnologies, ZAMM 92 (2012).