Vortex line

Vortex line (blue) with vortex vectors (black) of fluid elements (red)

The vortex line is a term from fluid mechanics related to the streamline : just as the velocity vector is tangential to the streamline, the vortex strength is tangential to the vortex line; a vortex line is thus affected at every point by the vortex strength, see picture. The vortex strength describes the rotation of the fluid elements around themselves by pointing in the direction of the axis of rotation of the elements and their amount is a measure of the rotational speed of the fluid elements.

A vortex surface is a surface formed by vortex lines in the flow, and a vortex tube is a tubular area, the outer surface of which consists of vortex lines. A vortex filament is - analogously to the current thread - a vortex tube having ( infinitesimal ) small cross-section, so that the fluid in the vortex filament properties as can be assumed to be constant over the cross section.

definition

A central variable in the theoretical description of vortices is the vortex strength , vortex density or the vortex vector

${\displaystyle {\vec {\omega }}:=\operatorname {rot} \,{\vec {v}}\,,}$

d. H. the "red" rotation of the speed field . ${\displaystyle {\vec {v}}}$

Occasionally it is also defined

${\displaystyle {\vec {\omega }}:={\frac {1}{2}}\operatorname {rot} {\vec {v}}}$,

which makes no significant difference.

Analogously to the streamline, the vortex line is defined using the differential equation ${\displaystyle {\vec {x}}_{\omega }(s)}$

${\displaystyle {\frac {\mathrm {d} {\vec {x}}_{\omega }}{\mathrm {d} s}}={\vec {\omega }}({\vec {x}}_{\omega }(s),t)\quad \Rightarrow \quad \mathrm {d} {\vec {x}}_{\omega }\parallel {\vec {\omega }}}$

with a curve parameter . ${\displaystyle s}$

properties

Eddy currents (vortices) in a glass

Smoke rings

The vortex theorems describe properties of vortex lines and vortex tubes in flows barotropic fluids with negligible viscosity .

According to Helmholtz's theorems of vortices, vortex lines are material lines, which means that vortex lines swim with the flow. Fluid elements that are on a vortex line remain attached to it.

Vortex tubes, the outer surface of which is formed from vortex lines, cannot end in the flowing fluid body: They therefore form rings, end at the edges of the flow area or are literally infinite. This property explains

• why by agitators tend excited vortex form through the entire fluid reaching vortex tubes, see upper image, and
• why smoke rings are remarkably stable, see picture below.

See also

• Kelvin's vortex law

literature

• Herbert Oertel Jr. (Ed.): Prandtl guide through fluid mechanics. 10th, completely revised edition. Vieweg, Braunschweig et al. 2001, ISBN 3-528-38209-0 .

Web links

• NA Adams: Fluid Mechanics 2. Introduction to the dynamics of fluids. (PDF) Lecture notes winter semester 2014/2015. Chair of Aerodynamics and Fluid Mechanics, Technical University of Munich, January 9, 2015, accessed on August 29, 2015 .