Vortex dynamics

Vortex dynamics

In 1858 Hermann von Helmholtz published his seminal paper entitled "Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen," in "Journal für die reine und angewandte Mathematik", vol. 55, pp.25-55. So important was the paper that a few years later P. G. Tait published an English translation, "On integrals of the hydrodynamical equations which express vortex motion", in "Philosophical Magazine", vol. 33, pp.485-512 (1867). In his paper Helmholtz established his three "laws of vortex motion" in much the same way one finds them in any advanced textbook of fluid mechanics today. For the next century or so vortex dynamics matured as a subfield of fluid mechanics, always commanding at least a major chapter in treatises on the subject. Thus, H. Lamb's well known "Hydrodynamics" (6th ed., 1932) devotes a full chapter to vorticity and vortex dynamics as does G. K. Batchelor's " [http://www.cambridge.org/uk/series/sSeries.asp?code=CML&srt=T Introduction to Fluid Dynamics] " (1967). In due course entire treatises were devoted to vortex motion. H. Poincaré's "Théorie des Tourbillons" (1893), [http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=80970 H. Villat's] "Leçons sur la Théorie des Tourbillons" (1930), C. Truesdell's "The Kinematics of Vorticity" (1954), and P. G. Saffman's " [http://www.cambridge.org/us/catalogue/email.asp?isbn=0521477395 Vortex Dynamics] " (1992) may be mentioned. Early on individual sessions at scientific conferences were devoted to vortices, vortex motion, vortex dynamics and vortex flows. Later, entire meetings were devoted to the subject.

The range of applicability of Helmholtz's work grew to encompass atmospheric and oceanographic flows, to all branches of engineering and applied science and, ultimately, to superfluids (today including Bose-Einstein condensates). In modern fluid mechanics the role of vortex dynamics in explaining flow phenomena is firmly established. Well known vortices have acquired names and are regularly depicted in the popular media: hurricanes, tornadoes, waterspouts, aircraft trailing vortices (e.g., wingtip vortices), drainhole vortices (including the bathtub vortex), smoke rings, underwater bubble air rings, cavitation vortices behind ship propellers, and so on. In the technical literature a number of vortices that arise under special conditions also have names: the Kármán vortex street wake behind a bluff body, Taylor vortices between rotating cylinders, Görtler vortices in flow along a curved wall, etc.

Today, one can scarcely imagine an investigation in fluid mechanics that does not invoke the role of vorticity or vortices in some way. Nevertheless, while vortices and vortex motion are ubiquitous, vortex dynamics has retained a characteristic "flavor" deriving from its particle-based (Lagrangian) interpretation and from its frequently intuitive, "mechanistic" description of flow phenomena. For example, the entire process of blowing out a candle by a puff of air is readily explained by vortex dynamics but is much more complicated to explain using the usual primitive variables of fluid flow theory such as pressure and velocity. In particular, the speed of the vortex ring that propagates from the origin of the puff to the candle is only readily understood when the vortex motion is fully elucidated.

A curious diversion in the history of vortex dynamics is the vortex atom theory of William Thomson, later Lord Kelvin. His basic idea was that atoms were to be represented as vortex motions in the ether. This theory predated the quantum theory by several decades and because of the scientific standing of its originator received considerable attention. Many profound insights into vortex dynamics were generated during the pursuit of this theory. Other interesting corrollaries were the first counting of simple knots by P. G. Tait, today considered a pioneering effort in graph theory, topology and knot theory. Ultimately, Kelvin's vortex atom was seen to be wrong-headed but the many results in vortex dynamics that it precipitated have stood the test of time. Kelvin himself originated the notion of circulation and proved that in an inviscid fluid circulation around a material contour would be conserved. This profound result – singled out by Einstein as one of the most significant results of Kelvin's work – provided an early link between fluid dynamics and topology.

The history of vortex dynamics seems particularly rich in discoveries and re-discoveries of important results, because results obtained were entirely forgotten after their discovery and then were re-discovered decades later. Thus, the integrability of the problem of three point vortices on the plane was solved in the 1877 thesis of a young Swiss applied mathematician named Walter Gröbli. In spite of having been written in Göttingen in the general circle of scientists surrounding Helmholtz and Kirchhoff, and in spite of having been mentioned in Kirchhoff's well known lectures on theoretical physics and in other major texts such as Lamb's "Hydrodynamics", this solution was largely forgotten. A 1949 paper by the noted applied mathematician J. L. Synge created a brief revival, but Synge's paper was in turn forgotten. A quarter century later a 1975 paper by E. A. Novikov and a 1979 paper by H. Aref on chaotic advection finally brought this important earlier work to light. The subsequent elucidation of chaos in the four-vortex problem, and in the advection of a passive particle by three vortices, made Gröbli's work part of "modern science".

Another example of this kind is the so-called "localized induction approximation" (LIA) for three-dimensional vortex filament motion which gained favor in the mid-1960s through the work of Arms, Hama, Betchov and others, but turns out to date from the early years of the 20'th century in the work of Da Rios, a gifted student of the noted Italian mathematician T. Levi-Civita. Da Rios published his results in several forms but they were never assimilated into the fluid mechanics literature of his time. In 1972 H. Hasimoto used Da Rios' "intrinsic equations" (later re-discovered independently by R. Betchov) to show how the motion of a vortex filament under LIA could be related to the non-linear Schrödinger equation. This immediately made the problem part of "modern science" since it was then realized that vortex filaments can support solitary twist waves of large amplitude.

Vortex dynamics continues today as a vibrant subfield of fluid dynamics, commanding special attention at major scientific conferences and precipitating workshops and symposia that focus fully on the subject.

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