Helmholtz vortex theorems

Curved but eddy-free flow away from stall

Eddy currents (vortices) in a glass

Smoke rings

The Helmholtz vortex theorems of Hermann von Helmholtz provide information about the behavior of vortices in flows barotropic , frictionless fluids. Apart from hydrodynamic boundary layers, these assumptions fit well with flows of fluids with low viscosity . For real gases at low pressures and high temperatures, freedom from friction is a proven assumption. The naming of the vortex sentences is not uniform in the literature. The listing here follows NA Adams:

First Helmholtz vortex law: In the absence of vortex-stimulating external forces, vortex-free flow areas remain vortex-free. This theorem is also simply called the Helmholtz vortex law or the third Helmholtz vortex law .

Second Helmholtz vortex law: Fluid elements that lie on a vortex line remain on this vortex line. Vortex lines are therefore material lines.

Third Helmholtz vortex law: The circulation along a vortex tube is constant. A vortex line therefore cannot end in the fluid. Vortex lines are - like streamlines in divergence-free flows - closed, literally infinite or run towards the edge. This theorem is also known as the first Helmholtz vortex law.

Even if the prerequisites of the vortex laws are only approximate in real flows, they explain

why eddies in laminar flows with many curves but not in a circle do not arise easily (without boundary layer effects such as flow breaks ), see picture above,

why by agitators tend excited vortices to be formed by all of the fluid reaching vortex tubes, see middle image, and

why smoke rings are remarkably stable, see picture below.

requirements

The vortex density or the vortex vector is a central variable in the theoretical description of vortices

{\ displaystyle {\ vec {\ omega}}: = \ operatorname {red} \, {\ vec {v}} \ ,,}

which is the rotation of the velocity field . Occasionally, a bet is also made , which does not make a significant difference. ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {\ omega}} = {\ frac {1} {2}} \ operatorname {red} {\ vec {v}}}$

The vortex line is analogous to the streamline 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}}}

defined with a curve parameter s. Just as the velocity vector is tangential to the streamline , the vortex vector is tangential to the vortex line. A vortex surface is a surface in the flow formed by vortex lines and a vortex tube is a tubular area whose surface area 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.

The Kelvin's theorem of vertebrae was historically formulated according to the Helmholtz 's theorem of vertebrae , but today it serves to prove the latter. It reads:

In the flow of a barotropic , frictionless fluid in a conservative gravitational field, the circulation Γ of the velocity around a closed, material curve b with a vector line element is constant over time: ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle \ mathrm {d} {\ vec {b}}}$

{\ displaystyle \ Gamma: = \ oint _ {b} {\ vec {v}} \ cdot \ mathrm {d} {\ vec {b}} \ quad \ Rightarrow \ quad {\ dot {\ Gamma}} = { \ frac {\ mathrm {D} \ Gamma} {\ mathrm {D} t}} = 0 \ ,.}

The differential operator and the dot stand for the substantial time derivative . The area integral of the vortex density over any area enclosed by the curve b is called the intensity of the vortex tube , which has the cross-sectional area a, and according to Stokes' integral theorem is equal to the circulation of the velocity along the curve b. The intensity of the vortex tube is also the same for all times. Only the third Helmholtz vortex law shows that a vortex tube only has an intensity that is constant over its entire length. ${\ displaystyle {\ tfrac {\ mathrm {D}} {\ mathrm {D} t}}}$

First Helmholtz vortex law

Helmholtz's first theorem of vortices states that areas free from vortices remain free from vortices in ideal liquids.

For the proof, a curve that encloses an (infinitesimal) small area a is placed around a rotation-free fluid element. Because of the small size, a vortex density that is constant over the area and disappears according to the assumption can be assumed, whose area integral is the intensity of the vortex tube with cross-sectional area a and this intensity also disappears according to the assumption. According to Kelvin's vortex law, the intensity is a conservation quantity, so that the vortex density disappears for all times in area a and therefore also for the fluid element under consideration.

Proof without Kelvin's vortex law
Forming the rotation in the Euler equations yields: ${\ displaystyle {\ begin {aligned} {\ frac {\ partial} {\ partial t}} \ operatorname {red} ({\ vec {v}}) + \ operatorname {red (grad} ({\ vec {v }}) \ cdot {\ vec {v}}) + {\ frac {1} {\ rho}} \ underbrace {\ operatorname {red (grad} (p))} _ {= {\ vec {0}} } = & {\ frac {\ partial} {\ partial t}} \ operatorname {red} ({\ vec {v}}) + \ underbrace {\ operatorname {red} \ left ({\ frac {1} {2 }} \ operatorname {grad} ({\ vec {v}} \ cdot {\ vec {v}}) \ right)} _ {= {\ vec {0}}} - \ operatorname {red} ({\ vec {v}} \ times \ operatorname {rot} {\ vec {v}}) = \ operatorname {red} ({\ vec {k}}) \\\ rightarrow {\ frac {\ partial} {\ partial t} } \ operatorname {red} ({\ vec {v}}) = & \ operatorname {red} ({\ vec {v}} \ times \ operatorname {red} ({\ vec {v}})) + \ operatorname {red} ({\ vec {k}}) \,. \ end {aligned}}}$ Here, the Grassmann development was used and made use of the fact that gradient fields are always free of rotation. The arithmetic symbol calculates the cross product . In a rotation-free gravitational field and with the vortex vector, the last equation follows: Development of the double cross product on the right-hand side yields, taking into account This is a linear differential equation of the first order for in . So if ever is anywhere , then the vortex vector must always vanish at the location of the associated mass element. ${\ displaystyle \ operatorname {grad} ({\ vec {v}}) \ cdot {\ vec {v}} = {\ frac {1} {2}} \ operatorname {grad} ({\ vec {v}} \ cdot {\ vec {v}}) - {\ vec {v}} \ times \ operatorname {red} ({\ vec {v}})}$ ${\ displaystyle \ times}$ ${\ displaystyle \ operatorname {red} ({\ vec {k}}) = {\ vec {0}}}$ ${\ displaystyle {\ frac {\ partial {\ vec {\ omega}}} {\ partial t}} = \ operatorname {red} ({\ vec {v}} \ times {\ vec {\ omega}}) \ ,.}$ ${\ displaystyle \ operatorname {div} {\ vec {v}} = \ operatorname {div} {\ vec {\ omega}} = 0}$ ${\ displaystyle {\ frac {\ mathrm {D {\ vec {\ omega}}}} {\ mathrm {D} t}} = \ operatorname {grad} ({\ vec {v}}) \ cdot {\ vec {\ omega}}}$ ${\ displaystyle {\ vec {\ omega}}}$ ${\ displaystyle t}$ ${\ displaystyle {\ vec {\ omega}} = {\ vec {0}}}$

In laminar flows , eddies do not necessarily arise if the flow is curved. The creation and destruction of eddies in a homogeneous fluid requires friction ( viscosity ) in the fluid.

Second Helmholtz vortex law

The second Helmholtz vortex law states that fluid elements that belong to a vortex line at any point in time remain on this vortex line for all time, which therefore moves with the fluid and is therefore a material line.

To prove this, a vortex surface is considered whose normal vector is by definition everywhere perpendicular to the vortex density. A closed curve cuts an area out of the vertebral surface. The area integral of the rotation of the speed, the vortex density, thus disappears over the area and is equal to the circulation of the speed along the curve lying in the vortex surface. According to Kelvin's law of vortices, the circulation of this curve, understood as material , remains constant zero, which is why the fluid elements along the curve remain on the vortex surface. A vortex line can be defined as the intersection of two vortex surfaces. Because the fluid elements along this vortex line are bound to both vortex surfaces simultaneously, the fluid elements must remain on the vortex line.

Third Helmholtz vortex law

Piece of a vortex tube (red) with cross-sectional areas a and b and lateral area m

Helmholtz's third law of vortices states that the circulation along a vortex tube is constant.

To prove this, a finitely long piece of a vortex tube is mentally cut out, which is bordered by two cross-sectional areas a and b and by a surface m between the two cross-sections, see picture. The Gaussian integral theorem is applied to the finite volume v of the vortex tube piece :

{\ displaystyle 0 = \ int _ {v} \ operatorname {div (red} ({\ vec {v}})) \, \ mathrm {d} v = \ int _ {v} \ operatorname {div} ({ \ vec {\ omega}}) \, \ mathrm {d} v = \ int _ {a} {\ vec {\ omega}} \ cdot \, \ mathrm {d} {\ vec {a}} _ {a }. + \ int _ {b} {\ vec {\ omega}} \ cdot \, \ mathrm {d} {\ vec {a}} _ {b} + \ int _ {m} {\ vec {\ omega }} \ cdot \, \ mathrm {d} {\ vec {a}} _ {m} \ ,.}

The differentials are the vectorial, outwardly directed surface elements of the surfaces a , b and m . Along the outer surface m of the vortex tube, the vortex vector is by definition parallel to the surface, so that the scalar product with the vectorial surface element disappears and the entire outer surface does not contribute anything to the above sum, i.e.: ${\ displaystyle \ mathrm {d} {\ vec {a}} _ {a}, \, \ mathrm {d} {\ vec {a}} _ {b}, \, \ mathrm {d} {\ vec { at the}}$ ${\ displaystyle \ mathrm {d} {\ vec {a}} _ {m}}$

{\ displaystyle \ int _ {a} {\ vec {\ omega}} \ cdot \, \ mathrm {d} {\ vec {a}} _ {a} = - \ int _ {b} {\ vec {\ omega}} \ cdot \, \ mathrm {d} {\ vec {a}} _ {b} \ ,.}

The vectorial surface elements on the cross-sectional areas a and b are oriented outwards and therefore directed opposite one another. If one of the two cross-sectional areas is reoriented, its area integral changes sign and the intensities of the vortex tube on both cross-sectional areas are found to be identical. The intensities are, however, equal to the circulations, from which the proposition follows.

Vortex tubes can neither begin nor end in the fluid and are therefore - like the streamlines in divergence-free flows - closed, literally infinite or run towards the edge. If the vertebral tube constricts locally, then the vertebral density must increase at this point.

The second and third Helmholtz vortex laws establish the remarkable stability of smoke rings . In reality, however, smoke rings will decay after a finite time due to dissipation, which is not taken into account in the proof of the theorems. The vortex tube, which is generated by a whisk and extends through the entire body of water, also disappears after a while after switching off the kitchen appliance for the same reason.

literature

M. Bestehorn: hydrodynamics and structure formation . Springer, 2006, ISBN 978-3-540-33796-6 .
MJ Lighthill: An Informal Introduction to Theoretical Fluid Mechanics . Oxford University Press, 1986, ISBN 0-19-853630-5 .
PG Saffman: Vortex Dynamics . Cambridge University Press, 1995, ISBN 0-521-42058-X .
AM Kuethe, JD Schetzer: Foundations of Aerodynamics . John Wiley & Sons, Inc., New York 1959, ISBN 0-471-50952-3 .

Individual evidence

^ NA Adams: Fluid Mechanics 2. Introduction to the dynamics of fluids . 2015 ( online [PDF; accessed August 29, 2015]).
^ M. Bestehorn: hydrodynamics and structure formation. 2006, p. 79.

Footnotes

↑ Currents flowing in circles are often eddies.

[1] NA Adams: Fluid Mechanics 2. Introduction to the dynamics of fluids . 2015 ( online [PDF; accessed August 29, 2015]).

[3] M. Bestehorn: hydrodynamics and structure formation. 2006, p. 79.

[2] Currents flowing in circles are often eddies.