Theodore von Kármán

How to use this course

Theodore von Kármán was trained as an engineer and thought like an applied mathematician. His gift was to find the one simplified model of a messy problem that could be solved and that still captured the physics: a pair of vortex rows for the wake of a cylinder, a chain of springs for a crystal, a similarity hypothesis for turbulence, a mass balance for a rocket motor. He was also a builder of institutions on three continents. He directed the aeronautical institute at Aachen, then the Guggenheim laboratory at Caltech, co-founded the Jet Propulsion Laboratory and the Aerojet company, chaired the US Air Force's scientific advisers, and founded NATO's aeronautical research group.

The course follows both strands. Part I covers his formation in Budapest and Göttingen and his years at Aachen. Part II presents his main contributions to the physics of solids and fluids. Part III treats his American career, his advisory work and his legacy. The page on his teacher Ludwig Prandtl is a natural companion; several results here extend those of Prandtl's course.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons run roughly in the order of his work: structures (1908–1932), the vortex street (1911–12), lattice dynamics (1912), laminar boundary layers (1921), turbulence (1921–1948), compressible and supersonic flow (1932–1956), and rockets and aeroelasticity (1940–41).

Timeline

Budapest, 1881–1906

Theodore von Kármán was born Kármán Tódor in Budapest on 11 May 1881, into a Jewish family. His father, Mór Kármán, was a professor of philosophy and education at the University of Budapest and a reformer of Hungarian secondary schooling; he was ennobled in 1907, which gave the family the prefix "von" in German. His mother, Helene Konn, came from a family of scholars. Theodore was one of several children; his sister Josephine later kept house for him in Aachen and Pasadena.

At about six he could multiply large numbers in his head. His father, worried that he would become a performing prodigy, steered him away from mathematics for several years and had him taught at home in geography, history and poetry. At nine he entered the Minta Gymnasium, the model school his father had founded on his own educational principles, where pupils were led to derive rules from examples rather than memorize them. In his final year he won the Eötvös Prize, awarded by competition to the best student in mathematics and science in Hungary.

Engineering in Budapest

His father pushed him toward a practical profession, and he studied mechanical engineering at the Royal Joseph Polytechnic in Budapest, graduating in 1902. His diploma work analysed the motion of a heavy rod resting on a rounded end on a horizontal plane. After a year as an artillery cadet he became an assistant in hydraulics at the Polytechnic and consulted for the Ganz engineering works. In 1906 he published a paper on the buckling of long slender columns, and the same year a two-year fellowship from the Hungarian Academy of Sciences took him to Göttingen.

Göttingen, 1906–1912

Göttingen in 1906 was the centre of world mathematics. Felix Klein was building applied science around it, and David Hilbert and Hermann Minkowski lectured there. Ludwig Prandtl had arrived two years earlier with the boundary-layer idea. Kármán joined Prandtl's group. His doctoral work, completed in 1908, combined theory and experiment on the buckling of columns stressed beyond the elastic limit, using a large hydraulic press (Lesson 04). The thesis was published in 1910 as Untersuchungen über Knickfestigkeit.

In March 1908 he made a short trip to Paris and watched one of the early powered flights near the city. The experience redirected his interests toward aeronautics. He habilitated at Göttingen in 1909 and taught there as a Privatdozent, lecturing on mechanics and thermodynamics.

Two famous papers

Kármán's best-known Göttingen story concerns a doctoral student of Prandtl, Karl Hiemenz, who was trying to measure the pressure distribution on a cylinder in a water channel. The flow behind the cylinder kept oscillating, and Hiemenz blamed his apparatus. Kármán suspected the oscillation was real and asked why the wake should prefer a staggered pattern of vortices. He calculated that symmetric vortex rows are always unstable and that staggered rows are stable only at one spacing ratio. The papers of 1911 and 1912 (the second with Henri Rubach) made the "Kármán vortex street" a standard object of fluid mechanics (Lesson 05).

The second paper took him into physics. With Max Born, then also a Privatdozent at Göttingen, he wrote in 1912 a theory of the vibrations of a crystal lattice that introduced the periodic boundary conditions now used throughout solid-state physics (Lesson 06). In the same period he wrote, for Klein's Encyklopädie der mathematischen Wissenschaften, a survey of strength problems in machine design containing the nonlinear plate equations now named after August Föppl and himself.

In 1912 he accepted a chair of applied mechanics at the mining academy of Selmecbánya (now Banská Štiavnica) in Hungary. He found it poorly equipped and returned to Göttingen after one term.

Aachen, war and revolution, 1913–1929

In February 1913 the Technische Hochschule in Aachen appointed him professor of aeronautics and mechanics and director of its Aeronautical Institute. He built up equipment, including a wind tunnel, and a research group. The First World War interrupted the work. From 1914 he served in the Austro-Hungarian army. At the military aviation centre at Fischamend near Vienna he worked on aircraft and propellers, and with István Petróczy and Vilém Žurovec he took part in building a tethered helicopter intended to replace observation balloons; prototypes were built and flown in 1917–18.

At the end of the war he was in Budapest. During the revolutionary months of 1918–19 he headed the department of technical higher education in the ministry of education and worked on university reform. When the Hungarian Soviet Republic fell in August 1919 he left Hungary, and by the end of the year he was back in Aachen with his mother and his sister Josephine.

The Aachen school

The 1920s were the years in which Kármán became an international figure. Aachen students built competitive gliders for the Rhön contests, and Wolfgang Klemperer, who designed them, took his doctorate with Kármán. In 1921 Kármán published his paper on laminar and turbulent friction (Lessons 07 and 08). In September 1922 he organized at Innsbruck the first international congress of applied mechanics, which led to the International Union of Theoretical and Applied Mechanics. German aerodynamics now had two centres, Göttingen and Aachen, and the friendly rivalry between Prandtl and his former student, especially over turbulence, sharpened both.

In 1926 Robert Millikan invited him to Caltech to advise on the new Guggenheim wind tunnel. Over the next years he divided his time between Aachen and Pasadena. Rising antisemitism in Germany made the decision easier, and in 1930 he moved to Caltech full time, with his mother and sister.

Buckling, plates and shells

For the general reader. A column pressed from both ends stays straight until a critical load, then suddenly bows sideways. Euler computed that load in the eighteenth century for perfectly elastic columns. Real columns made of steel often fail after the metal has started to yield, where Euler's formula overestimates their strength. Kármán's thesis measured and explained this inelastic buckling. He went on to write the equations that govern thin plates with large deflections, and later to explain why a buckled thin sheet in an aircraft can still carry load.

Elastic and inelastic columns

A pinned column of length \(L\), Young's modulus \(E\) and second moment of area \(I\), under axial load \(P\), has lateral deflection \(w(x)\) with \(EIw''+Pw=0\), \(w(0)=w(L)=0\). A non-zero solution \(w=\sin(\pi x/L)\) exists first at the Euler load \(P_E=\pi^2EI/L^2\).

Above the proportional limit the stress–strain curve flattens, with tangent modulus \(E_t=d\sigma/d\varepsilon<E\). Friedrich Engesser proposed in 1889 to replace \(E\) by \(E_t\). He then noted that during bending one side of the column unloads elastically, with modulus \(E\), while the other side loads with modulus \(E_t\). Kármán worked out this double-modulus theory independently in his thesis and tested it on carefully made steel columns.

Sketch. In the buckled state the neutral axis shifts toward the unloading side, to the place where the resultant of the extra stresses vanishes. For a rectangle of depth \(d\), let the unloading part have depth \(d_1\) and the loading part \(d_2\), \(d_1+d_2=d\). Force balance gives \(Ed_1^2=E_td_2^2\), so \(d_1/d_2=\sqrt{E_t/E}\). The bending moment is then \(M=-E_rI\,w''\) with \(E_r=4(Ed_1^3+E_td_2^3)/d^3\), which simplifies to the formula above (Exercise 4.1). In 1947 F. R. Shanley showed that a real column starts to bend at the lower tangent-modulus load \(P_t=\pi^2E_tI/L^2\), with increasing load, and that its maximum load lies between \(P_t\) and \(P_r\). Kármán accepted the refinement.

The Föppl–von Kármán equations (1910)

A thin elastic plate of thickness \(h\) whose deflection \(w\) is comparable to \(h\) stretches in its own plane as it bends. Let \(\Phi\) be the Airy stress function of the in-plane (membrane) stresses, \(\sigma_{xx}=\Phi_{yy}\), \(\sigma_{yy}=\Phi_{xx}\), \(\sigma_{xy}=-\Phi_{xy}\), and let \(D=Eh^3/[12(1-\nu^2)]\) be the bending stiffness.

Linearizing about a flat plate under prescribed in-plane compression gives the classical buckling problem. For a simply supported plate of width \(b\), compressed along its length, the critical stress is

with \(k=4\) for long plates. The nonlinear equations explain what happens next. Unlike a column, a buckled plate keeps carrying extra load, because the edges, held straight by the stiffeners, take up the stress.

The second form is Kármán's argument: since \(\sigma_{cr}\propto 1/b^2\), the width at which a plate buckles at \(\sigma_e\) is \(b\sqrt{\sigma_{cr}/\sigma_e}\). The formula became the basis of stressed-skin aircraft design.

The vortex street

For the general reader. Wind blowing past a wire or a chimney sheds swirls alternately from each side. They form two rows behind the body, with each swirl in one row opposite a gap in the other. The shedding makes telegraph wires sing and can shake structures. Kármán asked why the swirls choose this staggered pattern. His answer was a stability calculation: of all arrangements of two rows, only the staggered one with a particular ratio of width to spacing can persist.

Rows of vortices

A point vortex of circulation \(\Gamma\) at the origin has complex potential \(W=\frac{\Gamma}{2\pi i}\log z\). An infinite row of such vortices at \(z=nl\), \(n\in\mathbb Z\), has \(W=\frac{\Gamma}{2\pi i}\log\sin\frac{\pi z}{l}\), so the velocity it induces at a point \(z\) is

A single row induces no velocity on its own members, by symmetry. Now add a second row of vortices of circulation \(-\Gamma\), displaced by a vertical distance \(h\). In the symmetric arrangement each vortex faces one of the other row; in the staggered arrangement the second row is also shifted by \(l/2\). In both cases the whole pattern translates rigidly, with speed

relative to the fluid at infinity. Mallock (1907) and Henri Bénard (1908) had observed such wakes; Kármán supplied the dynamics.

Method. Displace the \(n\)th vortex of each row by small amounts proportional to \(e^{in\varphi}\), \(0\le\varphi\le2\pi\), and linearize the induced velocities. Each \(\varphi\) gives a \(4\times4\) linear system. Its eigenvalues come in pairs \(\pm\lambda\), with \(\lambda^2\) equal to \(\Gamma^2/l^4\) times an explicit function of \(\varphi\) and \(h/l\). Neutral stability requires \(\lambda^2\le0\) for every \(\varphi\). For the staggered street this happens only at \(\cosh(\pi h/l)=\sqrt2\). The result is subtle: a finer analysis shows that even this street is unstable to some nonlinear and finite-core perturbations. Its success lies elsewhere. The measured spacing ratio in real wakes is close to 0.28, and the model gave a first theory of the drag of bluff bodies from the momentum carried away by the vortices.

The Strouhal number

In 1878 Vincenc Strouhal found that the pitch of a wire singing in the wind is proportional to the wind speed divided by the wire diameter. In dimensionless form, the shedding frequency \(f\) behind a body of width \(D\) in a stream \(U\) satisfies

for a circular cylinder over a wide range of Reynolds numbers. Regular shedding begins at \(\mathrm{Re}=UD/\nu\approx47\), when the steady wake loses stability; above \(\mathrm{Re}\approx190\) the wake becomes three-dimensional, and near \(\mathrm{Re}\sim10^5\) shedding becomes irregular.

Lattice vibrations with Born

For the general reader. A crystal is a regular array of atoms held together by forces that act like springs. Heat makes the atoms vibrate, and the vibrations travel through the crystal as waves. In 1912 Max Born and Kármán computed these waves for a real lattice and used them to explain how the heat capacity of a solid falls toward zero at low temperature. Their trick of wrapping a finite crystal around on itself, so that it has no surface, is still used in almost every calculation in solid-state physics.

The problem in 1912

Einstein had explained in 1907 that the heat capacity of solids falls at low temperature because atomic vibrations are quantized. His model gave every atom the same frequency, and it predicted an exponential fall that experiment did not show. In 1912 Peter Debye treated the crystal as an elastic continuum with a cut-off frequency. Born and Kármán, in the same year, treated the actual lattice of atoms.

The monatomic chain

Consider \(N\) atoms of mass \(m\) on a line with spacing \(a\), nearest neighbours joined by springs of stiffness \(K\). With displacements \(u_n\),

A finite chain has ends, and the end atoms behave differently from the rest. Born and Kármán removed the ends by declaring \(u_{n+N}=u_n\): the chain is closed into a ring, or equivalently the crystal is repeated periodically.

For long waves, \(\omega\approx a\sqrt{K/m}\,|k|\): ordinary sound with speed \(c=a\sqrt{K/m}\). Near the zone boundary the waves become dispersive and their group velocity falls to zero. Each mode is a quantum oscillator, so the thermal energy is

with \(g(\omega)\) the density of states. Debye's model replaces \(g\) by its long-wave form up to a cut-off; the Born–von Kármán lattice gives the true \(g\), with sharp features (van Hove singularities) where the group velocity vanishes. At low temperature only long waves are excited, and both theories give a heat capacity \(C\propto T^3\) in three dimensions.

Boundary layers and the rotating disk

For the general reader. Prandtl's boundary-layer equations of 1904 were simpler than the full equations of viscous flow, but still hard to solve. In 1921 Kármán found a shortcut: instead of satisfying the equations at every point of the layer, satisfy only the balance of momentum for the layer as a whole. With a reasonable guess for the shape of the velocity profile, the problem reduces to one ordinary differential equation. Engineers used the method for forty years. In the same paper he solved exactly the flow next to a spinning disk, which pumps fluid like a fan.

The momentum integral

Integrating Prandtl's equation \(uu_x+vu_y=U_eU_e'+\nu u_{yy}\) across the layer, using continuity to eliminate \(v\), gives an exact relation between integral thicknesses and the wall stress \(\tau_w\).

Kármán's idea was to insert an assumed profile with one free parameter. In the same 1921 volume of the Zeitschrift für angewandte Mathematik und Mechanik, Ernst Pohlhausen used a quartic in \(\eta=y/\delta\),

chosen to satisfy the wall compatibility condition \(\nu u_{yy}(0)=-U_eU_e'\) and smooth matching at \(y=\delta\). The wall shear is proportional to \(2+\Lambda/6\), so the method predicts separation at \(\Lambda=-12\). The Kármán–Pohlhausen method gives the flat-plate drag within a few per cent and locates separation roughly; it was the standard engineering tool until computers made marching the full equations routine.

The rotating disk

An infinite disk at \(z=0\) rotates with angular velocity \(\Omega\) in fluid at rest far away. Kármán sought solutions in cylindrical coordinates \((r,\phi,z)\) of the form

The layer has constant thickness of order \(\sqrt{\nu/\Omega}\). Friction drags fluid round with the disk; centrifugal force throws it outward; and fluid is drawn axially toward the disk with speed \(0.884\sqrt{\nu\Omega}\) to replace it. It is an exact solution of the full Navier–Stokes equations with all three velocity components non-zero, and it models rotating machinery, viscometers and the boundary layer under a spinning fluid.

Turbulence

For the general reader. Turbulence was the problem Kármán returned to most often. He attacked it in three ways. First, with empirical laws and the momentum integral, he computed the drag of turbulent boundary layers for engineers. Second, he argued that turbulence near a wall looks the same at every scale, which leads to the logarithmic velocity profile and the constant that bears his name. Third, with Leslie Howarth, he wrote the first exact equation for the statistics of turbulence, the starting point of the modern theory.

The turbulent flat plate (1921)

Using Heinrich Blasius's empirical pipe-friction law, Prandtl and Kármán independently inferred that turbulent velocity profiles follow roughly \(u/U=(y/\delta)^{1/7}\), with wall stress \(\tau_w=0.0225\,\rho U^2(\nu/U\delta)^{1/4}\). Inserting these in the momentum integral gives, for a plate turbulent from its leading edge,

valid for \(\mathrm{Re}\) up to about \(10^7\). Fitted to plate measurements, the drag constant becomes 0.074.

The similarity hypothesis (1930)

Prandtl's mixing length \(\ell\) (see his course) had to be specified by hand. Kármán proposed that, away from the wall, turbulence is locally similar: the mean flow near each point is described by \(\bar u'\) and \(\bar u''\) alone, and the only length they define is \(\bar u'/\bar u''\). Hence

Derivation. Setting \(\rho\kappa^2(\bar u')^4/(\bar u'')^2=\tau_w\) gives \((\bar u')^2=-u_\tau\bar u''/\kappa\) for a profile with \(\bar u''<0\). Put \(\bar u'=p(y)\): \(p'=-\kappa p^2/u_\tau\), so \(1/p=\kappa y/u_\tau+c\). With \(c=0\) (the profile measured from the wall), \(\bar u'=u_\tau/(\kappa y)\) and the logarithm follows. Kármán's similarity length coincides with Prandtl's \(\ell=\kappa y\) for this profile, which is why the two men reached the same law at the same time and argued about priority. The number \(\kappa\) is the Kármán constant.

Isotropic turbulence (1938)

G. I. Taylor had introduced in 1935 the statistical theory of homogeneous isotropic turbulence. Kármán and Howarth showed that isotropy reduces the two-point velocity correlation to one scalar function. Let \(u'^2=\overline{u_1^2}\), and define the longitudinal correlations

The equation is exact but not closed: it links the double correlation to the unknown triple correlation \(k\), the closure problem in its simplest form. Kármán and Howarth looked for self-preserving solutions in which \(f\) keeps its shape as the turbulence decays. Andrey Kolmogorov derived his four-fifths law (1941) from the same equation, rewritten for structure functions. In 1948 Kármán proposed an interpolation for the energy spectrum, \(E(k)\propto k^4/(1+k^2/k_e^2)^{17/6}\), which behaves like \(k^4\) for large eddies and like Kolmogorov's \(k^{-5/3}\) in the inertial range; it is still used to generate synthetic turbulence and in atmospheric gust models.

High-speed aerodynamics

For the general reader. As aircraft approached the speed of sound in the 1930s, designers needed to know how compressibility changes the forces on a wing. Kármán and his student Qian Xuesen (Hsue-shen Tsien) produced a correction formula that worked better than Prandtl's simpler rule. Kármán also computed the wave drag of slender supersonic bodies such as shells and rockets, and found the scaling law that organizes transonic flight, the awkward regime just below and above Mach 1.

Subsonic compressibility (1939–41)

The Prandtl–Glauert rule \(C_p=C_{p,0}/\sqrt{1-M_\infty^2}\) comes from linearizing the potential equation. It underestimates suction peaks at higher subsonic Mach numbers. Tsien (1939) and Kármán (1941) used the hodograph method: in the plane of velocity components the equations are linear, and replacing the true adiabatic pressure–density curve by its tangent at free-stream conditions (formally \(\gamma=-1\)) makes them solvable in closed form.

For small \(C_{p,0}\) it reduces to Prandtl–Glauert; for suction (\(C_{p,0}<0\)) it predicts a larger correction, in better agreement with experiment up to the critical Mach number.

Supersonic slender bodies (1932)

Above Mach 1 the linearized equation \((M_\infty^2-1)\phi_{xx}=\phi_{yy}+\phi_{zz}\) is a wave equation, and disturbances are confined to Mach cones. Kármán and Norton Moore represented an axisymmetric body by a line of sources along its axis, the supersonic analogue of Rankine's method, and computed pressure and drag on projectiles.

The formula depends only on the area distribution, so two bodies with the same \(S(x)\) have the same wave drag. This is the root of the area rule found by Richard Whitcomb in 1952, and the shapes of least drag for given volume (Sears–Haack bodies) or given base area (the von Kármán ogive) come from minimizing it.

Transonic similarity (1947)

Near \(M_\infty=1\) the linear term \((1-M_\infty^2)\phi_{xx}\) is small, and the leading nonlinearity must be kept:

with \(\phi_y=U\tau g'(x)\) on a body of thickness ratio \(\tau\) and shape \(g\) (here \(M_\infty\approx1\) in the coefficient of the nonlinear term).

Sketch. Put \(y=Y/a\) and \(\phi=Ub\,\Phi(x,Y)\). The equation becomes \((1-M_\infty^2)b\Phi_{xx}-(\gamma+1)b^2\Phi_x\Phi_{xx}+a^2b\Phi_{YY}=0\) and the boundary condition \(ab\,\Phi_Y=\tau g'\). Choosing \(ab=\tau\) and \((\gamma+1)b=a^2\) removes \(\tau\) and \(\gamma\) from all but one coefficient: \(a=[(\gamma+1)\tau]^{1/3}\), \(b=\tau^{2/3}/(\gamma+1)^{1/3}\), and the remaining parameter is \((1-M_\infty^2)/a^2=K\). Then \(C_p=-2\phi_x/U=-2b\Phi_x\).

Rockets and structures in the wind

For the general reader. In the late 1930s rockets were the hobby of enthusiasts, and many engineers thought them unreliable. Solid-fuel rockets in particular tended either to fizzle or to explode. Kármán and his student Frank Malina showed with a short calculation that a solid motor can burn steadily, and the first American rocket-assisted take-off followed in 1941. In the same years Kármán was called to explain why the Tacoma Narrows Bridge had shaken itself apart in a moderate wind, a problem in what he called aeroelasticity.

The solid-propellant motor (1940)

Propellant of density \(\rho_p\) burns on a surface of area \(A_b\), receding at a rate that depends on chamber pressure, empirically \(r=ap^n\). Gas leaves through a nozzle throat of area \(A_t\) at the choked mass flow \(pA_t/c^*\), where the characteristic velocity \(c^*\) depends on the propellant chemistry. Malina and Kármán's analysis, in modern notation, is a mass balance in the chamber.

Stability. Including the gas mass \(m=pV/(RT)\) in the chamber, \(\dot m=\rho_pA_bap^n-pA_t/c^*\). For a small perturbation \(p=p_{eq}(1+\epsilon)\), the generation term grows like \(1+n\epsilon\) and the discharge like \(1+\epsilon\), so \(\dot\epsilon\propto(n-1)\epsilon\). With \(n<1\) a pressure rise increases discharge faster than generation and dies away. With \(n>1\) it feeds on itself, and the motor explodes. The GALCIT group's first jet-assisted take-off, on 12 August 1941 at March Field, used solid units built on this principle. In 1942 Jack Parsons devised a castable asphalt and perchlorate propellant, a forerunner of modern composite propellants.

Aeroelasticity and Tacoma Narrows (1940)

On 7 November 1940 the Tacoma Narrows suspension bridge twisted itself to destruction in a wind of about 19 m/s. Kármán, with Othmar Ammann and Glenn Woodruff, formed the board of engineers that reported on the failure in March 1941. Vortex shedding (Lesson 05) was the first suspect. Later analysis showed that the decisive motion was torsional flutter: once the deck began to twist, the aerodynamic moment acquired a component in phase with the angular velocity, acting as negative damping. A minimal model is

where \(c_s\) is the structural damping and \(c_a(U)\) the aerodynamic contribution, increasing with wind speed. Oscillations grow without bound when \(c_a(U)>c_s\). After the collapse, wind-tunnel testing of bridge models became standard practice, and Kármán, with his student Ernest Sechler and others at Caltech, developed aeroelasticity into a discipline for aircraft wings as well as bridges.

Caltech, JPL and the Air Force, 1930–1949

At Caltech Kármán directed the Guggenheim Aeronautical Laboratory (GALCIT) and its ten-foot wind tunnel. He gave American aeronautics what Göttingen had given Germany: theory joined to measurement. His seminars drew aircraft designers from the Los Angeles industry, and his doctoral students formed a generation of leaders. They included Maurice Biot (1932), Frank Wattendorf (1933), Ernest Sechler (1934), Qian Xuesen (1939), Frank Malina (1940), Homer Joe Stewart (1940), Chia-Chiao Lin (1944), Wallace Hayes (1947) and Frank Marble (1947). In 1933 he helped found the Institute of the Aeronautical Sciences. He was elected to the National Academy of Sciences in 1938 and gave the Gibbs Lecture of the American Mathematical Society in 1939.

From student rockets to JPL

In 1936 Malina, with Jack Parsons, Edward Forman and other students, asked to test rocket motors as a GALCIT project. Kármán gave them permission and his support, and the group fired its first motor in the Arroyo Seco above Pasadena that autumn. The Army Air Corps funded jet-assisted take-off research from 1939, the first American JATO flight took place on 12 August 1941, and in March 1942 Kármán, Malina, Parsons, Forman, Martin Summerfield and the lawyer Andrew Haley founded the Aerojet Engineering Corporation, with Kármán as president. In 1943 Kármán and his colleagues analysed intelligence on the German long-range rocket programme and proposed a missile research programme. In 1944 the Arroyo Seco group became the Jet Propulsion Laboratory, with Kármán as its first director; when he left for Washington at the end of that year, Malina succeeded him.

Adviser to the Air Force

In 1944 General Henry Arnold asked Kármán to chair a Scientific Advisory Group for the Army Air Forces. In 1945 Kármán led a mission to Europe that interviewed German scientists, including his old teacher Prandtl in Göttingen, and inspected wind tunnels and rocket sites. The group's multi-volume report, Toward New Horizons, set out a programme of research in jet propulsion, supersonic flight, guided missiles and electronics, and is often credited with shaping the post-war US Air Force's commitment to science. Kármán chaired the group's successor, the Air Force Scientific Advisory Board. He received the US Medal for Merit in 1946 and resigned his Caltech directorships in 1949.

Statesman of aeronautics, 1949–1963

After 1949 Kármán spent much of his time in Europe and on international organizations. He persuaded NATO to create the Advisory Group for Aeronautical Research and Development (AGARD), established in 1952, and chaired it until his death. In 1956 he founded near Brussels a training centre for experimental aerodynamics, renamed the von Karman Institute for Fluid Dynamics after his death; in the same year he helped found the International Council of the Aeronautical Sciences, and in 1960 the International Academy of Astronautics. His book Aerodynamics: Selected Topics in the Light of Their Historical Development (1954) remains a lucid introduction.

The postwar years also brought a bitter episode. His student Qian Xuesen, a founder of JPL and a colonel in the US Air Force during the 1945 mission, lost his security clearance in 1950 amid accusations of communist sympathies, was held under restriction for five years and was deported to China in 1955. There he led the Chinese missile and space programmes. Kármán supported him throughout.

In February 1963 President John F. Kennedy presented Kármán with the first National Medal of Science. He died on a visit to Aachen on 6 May 1963, five days before his 82nd birthday. (Some sources give 7 May.) He never married. His body was returned to the United States and entombed in the Beth Olam Mausoleum at what is now Hollywood Forever Cemetery in Los Angeles. His autobiography, The Wind and Beyond, written with Lee Edson, appeared in 1967.

Legacy

• Fluid mechanics: the vortex street, the momentum integral, the rotating-disk flow, the Kármán constant, the Kármán–Howarth equation and the von Kármán spectrum are standard textbook material.
• Solids and physics: the Föppl–von Kármán equations govern thin elastic sheets, from aircraft panels to crumpled paper and graphene; Born–von Kármán boundary conditions underlie band theory and phonon calculations.
• Aeronautics and space: compressibility corrections, transonic similarity and slender-body theory guided the design of high-speed aircraft; the Kármán line is the conventional edge of space.
• Institutions: the Jet Propulsion Laboratory, Aerojet, AGARD, the von Karman Institute and the international congresses of applied mechanics all carry his imprint.
• Honours: Foreign Member of the Royal Society (1946), Franklin Medal (1948), Daniel Guggenheim Medal (1955), first Ludwig-Prandtl-Ring (1957), first National Medal of Science (1963); craters on the Moon and Mars, and a 1992 US postage stamp.

Connections

People linked to Theodore von Kármán across this collection. A name without a link has no page here yet. See Theodore von Kármán in the network of all 337 lives →

Teachers and advisors

• Ludwig Prandtl 1906-1908

Students

• Chia-Chiao Lin 1944
• Ernest Sechler 1934
• Frank E. Marble 1947
• Frank Malina 1940
• Frank Wattendorf 1933
• Homer Joe Stewart 1940
• Maurice Anthony Biot 1932
• Qian Xuesen 1939
• Wallace D. Hayes 1947
• Wolfgang Klemperer 1926

Influenced by

• David Hilbert 1906-1912
• Felix Klein 1906-1912
• August Föppl

Influenced

• Andrey Kolmogorov

Colleagues and collaborators

• G. I. Taylor
• Max Born 1912
• Robert Millikan 1926-1949
• Tullio Levi-Civita 1922
• Jack Parsons
• Leslie Howarth 1938

Family

• Mór Kármán

Also mentioned in this life

• George Biddell Airy
• Jean-Baptiste Biot
• Peter Debye
• Albert Einstein
• Leonhard Euler
• Benjamin Franklin
• Josiah Willard Gibbs
• Ernst Mach
• Hermann Minkowski
• Claude-Louis Navier
• Max Planck
• Osborne Reynolds
• George Gabriel Stokes

Fields

• fluid mechanics 6 lives
• turbulence 5 lives
• solid mechanics 3 lives

Sources and further reading

Biography and history

1. J. J. O'Connor and E. F. Robertson, Theodore von Kármán, MacTutor History of Mathematics.
2. Theodore von Kármán, Wikipedia (doctoral students, honours).
3. Kármán, Theodor von, Neue Deutsche Biographie.
4. Toward New Horizons: Theodore von Kármán, in memoriam, Air Force Magazine, June 1963.
5. U.S. Centennial of Flight Commission, Early rocketry: the GALCIT project.
6. Kármán Tódor, Fizikai Szemle 2011, no. 7–8 (in Hungarian).
7. T. von Kármán with L. Edson, The Wind and Beyond, Little, Brown, 1967.
8. M. H. Gorn, The Universal Man: Theodore von Kármán's Life in Aeronautics, Smithsonian Institution Press, 1992.
9. H. L. Dryden, Theodore von Kármán 1881–1963, Biographical Memoirs of the National Academy of Sciences 38 (1965).

Physics

1. Th. von Kármán, Untersuchungen über Knickfestigkeit, Mitteilungen über Forschungsarbeiten des VDI 81 (1910).
2. Th. von Kármán, Festigkeitsprobleme im Maschinenbau, Encyklopädie der mathematischen Wissenschaften IV.4 (1910).
3. Th. von Kármán, Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt, Nachr. Ges. Wiss. Göttingen (1911, 1912).
4. M. Born and Th. von Kármán, Über Schwingungen in Raumgittern, Physikalische Zeitschrift 13 (1912).
5. Th. von Kármán, Über laminare und turbulente Reibung, ZAMM 1 (1921).
6. Th. von Kármán, Mechanische Ähnlichkeit und Turbulenz, Nachr. Ges. Wiss. Göttingen (1930).
7. Th. von Kármán, E. E. Sechler and L. H. Donnell, The strength of thin plates in compression, Trans. ASME 54 (1932).
8. Th. von Kármán and N. B. Moore, Resistance of slender bodies moving with supersonic velocities, Trans. ASME 54 (1932).
9. Th. von Kármán and L. Howarth, On the statistical theory of isotropic turbulence, Proc. R. Soc. Lond. A 164 (1938).
10. Th. von Kármán, Compressibility effects in aerodynamics, J. Aeronaut. Sci. 8 (1941).
11. Th. von Kármán, The similarity law of transonic flow, J. Math. Phys. 26 (1947).
12. Th. von Kármán, Aerodynamics: Selected Topics in the Light of Their Historical Development, Cornell University Press, 1954 (Dover reprint).
13. H. Schlichting and K. Gersten, Boundary-Layer Theory, 9th ed., Springer, 2017 (momentum integral, rotating disk, turbulent friction laws).
14. H. Lamb, Hydrodynamics, 6th ed., Cambridge, 1932 (vortex rows and the vortex street).
15. N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1976, chapters 8 and 22 (Born–von Kármán conditions, lattice dynamics).