Vortex strength

The vortex strength is a key parameter in fluid mechanics and meteorology by assigning a field of velocities to the vortex and the circular or spiral currents. The equivalent denomination vorticity from Latin vortex = "vortex, vortex," English vorticity is with Wirbelhaftigkeit translated.

In fluid mechanics, small differences in the speed and direction of gases and liquids are called shear . On the other hand, the streamlines are geometric aids for the descriptive description of a flow as a directed movement of particles. After all, viscosity is the viscosity of fluids , i.e. the resistance of the fluid to shear.

The vortex strength clearly corresponds to the tendency of a fluid element to rotate on its own around an axis, from which a circulation of flowing or flowing media arises in a closed area. Furthermore, the mean of the square vortex strength over a certain area is called enstrophy , which describes, for example, the flow behavior of double glass facades .

Formal notation

The vortex strength , referred to in meteorology based on the circulation , is defined as the rotation of the speed of a vector field : ${\displaystyle {\vec {\omega }}}$ ${\displaystyle {\vec {\zeta }}}$ ${\displaystyle {\vec {u}}}$

${\displaystyle {\vec {\omega }}:=\operatorname {rot} {\vec {u}}={\vec {\nabla }}\times {\vec {u}}}$

and is thus a pseudo- vector field, as is every rotation of a vector field , it has the SI unit . Because the conservation quantities do not change in a closed system, the vortex strength is equal to the area-related circulation rate : ${\displaystyle {\tfrac {1}{\mathrm {s} }}}$ ${\displaystyle {\vec {\omega }}}$ ${\displaystyle \Gamma }$

${\displaystyle \Gamma :=\oint _{\partial A}{\vec {v}}\cdot \mathrm {d} {\vec {r}}=\int _{A}\;\operatorname {rot} ({\vec {v}})\cdot \mathrm {d} {\vec {A}}}$

${\displaystyle \Rightarrow {\vec {\omega }}\cdot {\vec {n}}={\frac {\mathrm {d} \Gamma }{\mathrm {d} A}}}$

with the normal . In meteorology, except for genuinely three-dimensional eddies such as tornadoes , there are often two-dimensional velocity fields. The corresponding vorticity points in the z direction and reads ${\displaystyle {\vec {n}}}$

${\displaystyle {\vec {\zeta }}:=\operatorname {rot} {\vec {u}}_{2D}=\left({\frac {\partial u_{y}}{\partial x}}-{\frac {\partial u_{x}}{\partial y}}\right){\vec {e}}_{z}}$.

Hydrodynamics

In hydrodynamics , vorticity is the rotation of the fluid velocity, which is oriented in the direction of the axis of rotation or, for two-dimensional flows, perpendicular to the flow plane. For fluids with a fixed rotation around an axis (e.g. a rotating cylinder) the vortex strength is equal to twice the angular velocity  ω _{0 of} the fluid element:

${\displaystyle {\vec {\omega }}=2{\vec {\omega }}_{0}}$

${\displaystyle {\vec {\omega }}={\vec {\nabla }}\times {\vec {u}}={\vec {\nabla }}\times \left({\vec {\omega }}_{0}\times {\vec {r}}\right)={\vec {\omega }}_{0}\left({\vec {\nabla }}\cdot {\vec {r}}\right)-\left({\vec {\omega }}_{0}\cdot {\vec {\nabla }}\right){\vec {r}}=3{\vec {\omega }}_{0}-{\vec {\omega }}_{0}=2{\vec {\omega }}_{0}}$

Fluids without vorticity hot rotationally or wirbelfrei with . However, the fluid elements of such a non-rotating fluid can also have an angular velocity , i. H. move on curved paths, cf. the following figure, where the letter in the text stands for the vortex strength and in the figure for the angular velocity: ${\displaystyle {\vec {\omega }}=0\;{\text{bzw.}}\;\zeta =0}$ ${\displaystyle {\vec {\omega }}_{0}\neq 0}$ ${\displaystyle \omega }$

Vorticity and angular velocity

One looks at an infinitesimally small, square area of ​​a liquid. When this area rotates, the vortex strength of the flow is non-zero. The vortex strength refers to forced vortices with . ${\displaystyle {\vec {\omega }}=\operatorname {rot} {\vec {u}}\neq 0}$

Vorticity is a suitable means for liquids with low viscosity . Then the vorticity can be regarded as zero at almost all locations of the flow. This is obvious for two-dimensional flows, in which the flow can be represented on the complex plane . Such problems can usually be solved analytically.

For any flow, the governing equations can be related to the vortex strength instead of the velocity by simply substituting them. This leads to the eddy density equation , which for incompressible , non-viscous liquids is as follows:

${\displaystyle {\frac {\mathrm {D} {\vec {\omega }}}{\mathrm {D} t}}=({\vec {\omega }}\cdot {\vec {\nabla }}){\vec {u}}}$

Even for real flows (three-dimensional, finite Reynolds number , i.e. viscosity not equal to zero), the consideration of the flow via the vortex strength can be used with restrictions if one assumes that the vorticity field can be represented as an arrangement of individual vortices. The diffusion of these eddies through the flow is described by the vortex transport equation:

${\displaystyle {\frac {\mathrm {D} {\vec {\omega }}}{\mathrm {D} t}}=({\vec {\omega }}\cdot {\vec {\nabla }}){\vec {u}}+{\frac {\eta }{\rho }}\cdot (\nabla ^{2}{\vec {\omega }})}$

where denotes the Laplace operator . Here the eddy density equation was supplemented by the diffusion term . ${\displaystyle \nabla ^{2}}$ ${\displaystyle {\frac {\eta }{\rho }}\cdot (\nabla ^{2}{\vec {\omega }})}$

For highly viscous flows, for example Couette flows , it can make more sense to look directly at the velocity field of the fluid instead of the vortex strength, since the high viscosity leads to a very strong diffusion of the vortices.

The vortex line is directly related to the vortex strength in that the vortex lines are tangents to the vortex strength. The entirety of the vortex lines going through a surface element is called the vortex filament . The Helmholtz vortex theorems state that the vortex flow is constant in both time and space. ${\displaystyle dA}$ ${\displaystyle \iint {\vec {\omega }}\cdot \mathrm {d} {\vec {A}}}$

meteorology

In meteorology, vorticity is mainly used to describe the rotation of air around an axis. The absolute vorticity of a volume element or a body in meteorology is composed of two summands, the planetary vorticity and the relative . ${\displaystyle f_{\mathrm {c} }}$ ${\displaystyle \omega _{\mathrm {rel} }}$

Due to the rotation of the earth , every body close to the earth experiences a rotation around the earth's axis and thus has a fixed vorticity . This is determined by the Coriolis factor, which depends on the latitude ${\displaystyle \omega _{0}}$ ${\displaystyle f}$

${\displaystyle f_{\mathrm {c} }=2\omega _{0}\sin \varphi \approx 10^{-4}\,{\frac {1}{\mathrm {s} }}}$

determined and called planetary vorticity . The relative vorticity is the quantity related to the rotation of the body. When added, the absolute vorticity results :

${\displaystyle \omega _{\mathrm {abs} }=\omega _{\mathrm {rel} }+f_{\mathrm {c} }}$

Since mostly two-dimensional flow fields occur in meteorology, the relative vorticity is often expressed by the rotation in two dimensions

${\displaystyle {\vec {\zeta }}:=\operatorname {rot} {\vec {u}}_{2D}}$

This results in the absolute vorticity

${\displaystyle \omega _{\mathrm {abs} }=\colon \eta =\zeta +f}$

The direction of the vortex strength vector can be determined using the corkscrew rule: If the fluid rotates counter-clockwise, the vortex strength points upwards, which is positive in this case. In the northern hemisphere, counterclockwise rotation, i.e. with a positive rotation , is called cyclonic rotation and with a negative rotation, it is called anti- cyclonic rotation . In the southern hemisphere, the opposite applies accordingly. In natural coordinates we get: ${\displaystyle \eta }$ ${\displaystyle \eta }$ ${\displaystyle \eta }$

${\displaystyle \zeta =\zeta _{C}+\zeta _{S}=v\cdot K_{s}-{\frac {\partial v}{\partial n}}}$ With

${\displaystyle \zeta _{C}:=v\cdot K_{s}}$ as curvature vorticity and

${\displaystyle \zeta _{S}:=-{\frac {\partial v}{\partial n}}}$ as shear vorticity.

${\displaystyle K_{s}}$is the curvature of the streamlines , while n and s are the components of the coordinate system. Helmholtz's law of conservation for eddy flow leads to potential vorticity  PV. By combining the eddy density equation with the continuity equation , one can show that

${\displaystyle PV={\frac {\eta }{\Delta p}}={\frac {\zeta +f}{\Delta p}}}$

is preserved in time.

Remarks

The literature also contains the definition

${\displaystyle {\vec {\omega }}={\frac {1}{2}}\operatorname {rot} {\vec {u}}}$

The terms vorticity , vortex density , Wirbelhaftigkeit , vorticity , turbulence , vorticity , vortex filament , the appointment of Wirbeldichte- and vorticity equation are not clearly defined and therefore hard against each other defined. In the literature there are sometimes contradicting information and definitions.

literature

• Hans Stephani, Gerhard Kluge: Theoretical Mechanics . Spektrum Akademischer Verlag, Heidelberg 1995, ISBN 3-86025-284-4
• Ludwig Bergmann, Clemens Schaefer: Textbook of Experimental Physics . Volume 1: Mechanics, Relativity, Heat . de Gruyter, Berlin 1998. ISBN 3-11-012870-5
• Lew D. Landau, Jewgeni M. Lifschitz: Textbook of theoretical physics . Volume 6: hydrodynamics . Harri Deutsch publishing house, Frankfurt am Main 2007. ISBN 978-3-8171-1331-6
• Koji Ohkitani: Elementary Account Of Vorticity And Related Equations . Cambridge University Press, 2005. ISBN 0-521-81984-9
• Andrew J. Majda, Andrea L. Bertozzi: Vorticity and Incompressible Flow . Cambridge University Press, 2002. ISBN 0-521-63948-4

Individual evidence

1. ↑ Ludwig Bergmann, Clemens Schaefer: Textbook of Experimental Physics, Volume 1: Mechanics, Relativity, Warmth , p. 564. de Gruyter, Berlin 1998. ISBN 3-11-012870-5
2. ^ Roland Netz: Mechanics of the Continua. (PDF; 671 kB) Retrieved May 25, 2011 . 
3. ^ Vortex transport equations. (No longer available online.) Archived from the original on September 21, 2008 ; Retrieved May 25, 2011 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
4. ^ Vorticity. Retrieved May 25, 2011 . 
5. ^ Atmospheric physics . (PDF; 337 kB) (No longer available online.) Archived from the original on February 18, 2015 ; Retrieved May 25, 2011 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
6. ↑ scienceworld.wolfram.com. Retrieved May 25, 2011 . 
7. ↑ Hans Stephani, Gerhard Kluge: Theoretical Mechanics . Spektrum Akademischer Verlag, Heidelberg 1995, ISBN 3-86025-284-4 , p. 273.