Helmholtz's theorems

Helmholtz's theorems

In fluid mechanics, Helmholtz's theorems describe the three-dimensional motion of fluid in the vicinity of vortex filaments. These theorems apply to inviscid flows and flows where the influence of viscous forces is small and can be ignored.

Helmholtz’s three theorems are as follows: [Kuethe and Schetzer, "Foundations of Aerodynamics", Section 2.14]
Helmholtz’s first theorem:
:"The strength of a vortex filament is constant along its length."Helmholtz’s second theorem:
:"A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path."Helmholtz’s third theorem:
:"In the absence of rotational external forces, a fluid that is initially irrotational remains irrotational."

Helmholtz’s theorems apply to inviscid flows. In observations of vortices in real fluids the strength of the vortices always decays gradually due to the dissipative effect of viscous forces.

Alternative expressions of the three theorems are as follows:
1. The strength of a vortex tube does not vary with time. [The strength of a vortex tube (circulation), is defined as:: Gamma = int_{A} vec{omega} cdot vec{n} dA = oint_{c} vec{u} cdot dvec{s} where Gamma is also the circulation, vec{omega} is the vorticity vector, vec{n} is the normal vector to a surface A, formed by taking a cross-section of the vortex-tube with elemental area dA, vec{u} is the velocity vector on the closed curve C, which bounds the surface A. The convention for defining the sense of circulation and the normal to the surface A is given by the right-hand screw rule. The third theorem states that this strength is the same for all cross-sections A of the tube and is independent of time. This is equivalent to saying : frac{D Gamma}{Dt} = 0.]
2. Fluid elements lying on a vortex line at some instant continue to lie on that vortex line. More simply, vortex lines move with the fluid. Also vortex lines and tubes must appear as a closed loop, extend to infinity or start/end at solid boundaries.
3. Fluid elements initially free of vorticity remain free of vorticity.

Helmholtz’s theorems have application in understanding:

Generation of lift on an airfoil

Starting vortex

Horseshoe vortex

Wingtip vortices

Helmholtz’s theorems are now generally proven with reference to Kelvin's circulation theorem. However the Helmholtz's theorems were published in 1858, nine years before the 1867 publication of Kelvin's theorem. There was much communication between the two men on the subject of vortex lines, with many references to the application of their theorems to the study of smoke rings.Fact|date=May 2008

References

* M. J. Lighthill, "An Informal Introduction to Theoretical Fluid Mechanics", Oxford University Press, 1986, ISBN 0-19-853630-5
* P. G. Saffman, "Vortex Dynamics", Cambridge University Press, 1995, ISBN 0-521-42058-X
* G. K. Batchelor, "An Introduction to Fluid Dynamics", Cambridge University Press (1967, reprinted in 2000).
* Kundu, P and Cohen, I, "Fluid Mechanics", 2nd edition, Academic Press 2002.
* George B. Arfken and Hans J. Weber, "Mathematical Methods for Physicists", 4th edition, Academic Press: San Diego (1995) pp. 92-93
* A.M. Kuethe and J.D. Schetzer (1959), "Foundations of Aerodynamics", 2nd edition. John Wiley & Sons, Inc. New York ISBN 0 471 50952 3

Notes

ee also

Kelvin's circulation theorem

Vortex

Starting vortex

Horseshoe vortex

Wingtip vortices


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Helmholtz theorem — There exist several theorems named after Hermann von Helmholtz. * Helmholtz theorem in vector calculus , also known as fundamental theorem of vector calculus ; see Helmholtz decomposition. * Helmholtz theorem in classical mechanics ; see… …   Wikipedia

  • Helmholtz-Zerlegung — Das Helmholtz Theorem (auch Helmholtz Zerlegung) ist nach Hermann von Helmholtz benannt. Es besagt, dass für gewisse Gebiete der Lp Raum als direkte Summe von divergenzfreien Funktionen und Gradientenfeldern geschrieben werden kann.… …   Deutsch Wikipedia

  • Helmholtz-Theorem — Das Helmholtz Theorem, auch Helmholtz Zerlegung, (nach Hermann von Helmholtz) besagt, dass für gewisse Gebiete der Lp Raum als direkte Summe von divergenzfreien Funktionen und Gradientenfeldern geschrieben werden kann. Inhaltsverzeichnis 1… …   Deutsch Wikipedia

  • Sätze von Helmholtz — Das Helmholtz Theorem (auch Helmholtz Zerlegung) ist nach Hermann von Helmholtz benannt. Es besagt, dass für gewisse Gebiete der Lp Raum als direkte Summe von divergenzfreien Funktionen und Gradientenfeldern geschrieben werden kann.… …   Deutsch Wikipedia

  • Vortex — A vortex (pl. vortices ) is a spinning, often turbulent, flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex. The speed and rate of rotation of the… …   Wikipedia

  • List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia

  • Vorticity — is a mathematical concept used in fluid dynamics. It can be related to the amount of circulation or rotation (or more strictly, the local angular rate of rotation) in a fluid.Clancy, L.J., Aerodynamics , Section 7.11] The average vorticity zeta… …   Wikipedia

  • RIEDIGER — AUSTRIA (see also List of Individuals) 8.8.1880 Wien/A 20.10.1957 Wien/A Karl Riediger graduated as a civil engineer from Vienna Technical University in 1905 and moved than as a design engineer to Innsbruck in the County of Tyrol. He stayed in… …   Hydraulicians in Europe 1800-2000

  • GRAETZ — GERMANY (see also List of Individuals) 26.9.1856 Breslau/PL 12.11.1941 Munich/D Leo Graetz studied mathematics and physics at the Universities of Breslau, today s Wrocław in Poland, Berlin and Strasburg. From 1881 he was an assistant at Strasburg …   Hydraulicians in Europe 1800-2000

  • Fluid dynamics — Continuum mechanics …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”