Editing Vortex ring (section)

==Theory==

===Historical studies===
The formation of vortex rings has fascinated the scientific community for more than a century, starting with [[William Barton Rogers]]<ref>{{cite journal |last1= Rogers|first1= W. B. |date=1858 |title=On the formation of rotating rings by air and liquids under certain conditions of discharge |url=https://www.biodiversitylibrary.org/item/113539#page/255/mode/1up |journal=Am. J. Sci. Arts |volume=26 |pages=246–258 }}</ref> who made sounding observations of the formation process of air vortex rings in air, air rings in liquids, and liquid rings in liquids. In particular, [[William Barton Rogers]] made use of the simple experimental method of letting a drop of liquid fall on a free liquid surface; a falling colored drop of liquid, such as milk or dyed water, will inevitably form a vortex ring at the interface due to the surface tension.{{fact|date=April 2024}}

Vortex rings were first mathematically analyzed by the German physicist [[Hermann von Helmholtz]], in his 1858 paper ''On Integrals of the Hydrodynamical Equations which Express Vortex-motion''.<ref name="helmholtz1858">{{cite journal |last1=Helmholtz |first1=H. |date=1858 |title=3. Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen |url=https://ia800708.us.archive.org/view_archive.php?archive=/22/items/crossref-pre-1909-scholarly-works/10.1515%252Fcrll.1857.53.1.zip&file=10.1515%252Fcrll.1858.55.25.pdf |journal=Journal für die reine und angewandte Mathematik |volume=55 |pages=25–55 |doi=10.1515/9783112336489-003|isbn=9783112336472 }}</ref><ref name="helmholtz1867">{{cite journal |last1=Helmholtz |first1=H. |date=1867 |title=LXIII. On integrals of the hydrodynamical equations, which express vortex-motion |url=https://www.tandfonline.com/doi/pdf/10.1080/14786446708639824 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=33 |issue=226 |pages=485–512 |doi=10.1080/14786446708639824|url-access=subscription }}</ref><ref>{{cite book |last1=Moffatt |first1=K. |chapter=Vortex Dynamics: The Legacy of Helmholtz and Kelvin |date=2008 |editor1-last=Borisov |editor1-first=A. V. |editor2-last=Kozlov |editor2-first=V. V. |editor3-last=Mamaev |editor3-first=I. S. |editor4-last=Sokolovskiy |editor4-first=M. A. |title=IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence |chapter-url=https://link.springer.com/chapter/10.1007%2F978-1-4020-6744-0_1 |series=IUTAM Bookseries |publisher=Springer Netherlands |volume=6 |pages=1–10 |doi=10.1007/978-1-4020-6744-0_1|isbn=978-1-4020-6743-3 }}</ref>

===Circular vortex lines===
For a single zero-thickness vortex ring, the vorticity is represented by a [[Dirac delta function]] as <math> \omega\left(r,x\right)=\kappa\delta\left(r-r'\right)\delta\left(x-x'\right)</math> where <math> \left(r',x'\right)</math> denotes the coordinates of the vortex filament of strength <math>\kappa</math> in a constant <math>\theta</math> half-plane. The [[Stokes stream function]] is:<ref name="lamb1932">{{cite book|last1=Lamb|first1=H.|title=Hydrodynamics|publisher=Cambridge University Press|date=1932|pages=236–241|url=https://archive.org/details/hydrodynamics00lamb}}
<!-- minus sign missing, see Lamb p. 237 --></ref>
<math display="block"> \psi(r,x)=-\frac{\kappa}{2\pi}\left(r_1+r_2\right)\left[K(\lambda)-E(\lambda)\right] </math>
with <math> r_1^2 = \left(x-x'\right)^2+\left(r-r'\right)^2 \qquad r_2^2 = \left(x-x'\right)^2+\left(r+r'\right)^2 \qquad \lambda = \frac{r_2-r_1}{r_2+r_1} </math>
where <math>r_1</math> and <math>r_2</math> are respectively the least and the greatest distance from the point <math>P(r,x)</math> to the vortex line, and where <math>K</math> is the [[complete elliptic integral of the first kind]] and <math>E</math> is the [[complete elliptic integral of the second kind]].

A circular vortex line is the limiting case of a thin vortex ring. Because there is no core thickness, the speed of the ring is infinite, as well as the [[kinetic energy]]. The hydrodynamic impulse can be expressed in term of the strength, or 'circulation' <math>\kappa</math>, of the vortex ring as <math>I = \rho \pi \kappa R^2 </math>.

===Thin-core vortex rings===
The discontinuity introduced by the [[Dirac delta function]] prevents the computation of the speed and the [[kinetic energy]] of a circular vortex line. It is however possible to estimate these quantities for a vortex ring having a finite small thickness. For a thin vortex ring, the core can be approximated by a disk of radius <math>a</math> which is assumed to be infinitesimal compared to the radius of the ring <math>R</math>, i.e. <math>a/R \ll 1 </math>. As a consequence, inside and in the vicinity of the core ring, one may write: <math> r_1/r_2 \ll 1 </math>, <math>r_2 \approx 2R</math> and <math> 1- \lambda^2 \approx 4 r_1/R </math>, and, in the limit of <math>\lambda \approx 1 </math>, the elliptic integrals can be approximated by <math> K(\lambda) = 1/2 \ln\left({16}/{(1-\lambda^2)}\right) </math> and <math> E(\lambda) = 1 </math>.<ref name="lamb1932"/>

For a uniform [[vorticity]] distribution <math>\omega(r,x)=\omega_0</math> in the disk, the [[Stokes stream function]] can therefore be approximated by
<!-- minus sign missing and wrong bracket, see Lamb p. 241 --></ref>
<math display="block"> \psi(r,x)=-\frac{\omega_0}{2\pi}R\iint{\left(\ln\frac{8R}{r_1}-2\right)\,dr'dx'} </math>

The resulting [[circulation (physics)|circulation]] <math>\Gamma</math>, hydrodynamic impulse <math>I</math> and [[kinetic energy]] <math>E</math> are
<math display="block">\begin{align}
\Gamma &= \pi\omega_0 a^2\\
I &= \rho\pi\Gamma R^2 \\
E &= \frac{1}{2}\rho\Gamma^2R\left(\ln\frac{8R}{a}-\frac{7}{4}\right)
\end{align}</math>

It is also possible to find the translational ring speed (which is finite) of such isolated thin-core vortex ring:
<math display="block"> U=\frac{E}{2I}+\frac{3}{8\pi}\frac{\Gamma}{R} </math>
which finally results in the well-known expression found by [[Lord Kelvin|Kelvin]] and published in the English translation by [[Peter Tait (physicist)|Tait]] of [[von Helmholtz]]'s paper:<ref name="helmholtz1858"/><ref name="helmholtz1867"/><ref name="lamb1932"/>
<math display="block"> U=\frac{\Gamma}{4\pi R}\left(\ln\frac{8R}{a}-\frac{1}{4}\right) </math>

===Spherical vortices===
{{Main|Hill's spherical vortex}}

[[Micaiah John Muller Hill|Hill]]'s spherical vortex<ref name="hill1894">{{cite journal|last1=Hill|first1=M.J.M.|title=VI. On a spherical vortex |journal=Philosophical Transactions of the Royal Society of London A |date=1894 |volume=185 |pages=213–245 |doi=10.1098/rsta.1894.0006|bibcode=1894RSPTA.185..213H|doi-access=free }}</ref> is an example of steady vortex flow and may be used to model vortex rings having a vorticity distribution extending to the centerline. More precisely, the model supposes a linearly distributed vorticity distribution in the radial direction starting from the centerline and bounded by a sphere of radius <math>a</math> as:
<math display="block"> \frac{\omega}{r}=\frac{15}{2}\frac{U}{a^2}</math>
where <math>U</math> is the constant translational speed of the vortex.

Finally, the [[Stokes stream function]] of Hill's spherical vortex can be computed and is given by:<ref name="hill1894"/><ref name="lamb1932"/>
<math display="block">\begin{align}
&\psi(r,x) = -\frac{3}{4}\frac{U}{a^2}r^2\left(a^2-r^2-x^2\right) && \text{inside the vortex} \\
&\psi(r,x) = \frac{1}{2}Ur^2\left[1-\frac{a^3}{\left(x^2+r^2\right)^{3/2}}\right] && \text{outside the vortex}
\end{align}</math>
The above expressions correspond to the stream function describing a steady flow. In a fixed frame of reference, the stream function of the bulk flow having a speed <math>U</math> should be added.

The [[circulation (physics)|circulation]], the hydrodynamic impulse and the [[kinetic energy]] can also be calculated in terms of the translational speed <math>U</math> and radius <math>a</math>:<ref name="hill1894"/><ref name="lamb1932"/>
<!-- density ρ missing in impulse and energy , see Lamb p. 244 -->
<math display="block">\begin{align}
\Gamma &= 5Ua \\
I &= 2\pi\rho Ua^3 \\
E & = \frac{10\pi}{7}\rho U^2a^3
\end{align}</math>

Such a structure or an electromagnetic equivalent has been suggested as an explanation for the internal structure of [[ball lightning]].  For example, Shafranov {{Citation needed|date=July 2010}} used a magnetohydrodynamic (MHD) analogy to Hill's stationary fluid mechanical vortex to consider the equilibrium conditions of axially symmetric MHD configurations, reducing the problem to the theory of stationary flow of an incompressible fluid.  In axial symmetry, he considered general equilibrium for distributed currents and concluded under the [[Virial Theorem]] that if there were no gravitation, a bounded equilibrium configuration could exist only in the presence of an azimuthal current.{{Citation needed|date=June 2021}}

===Fraenkel-Norbury model===

The Fraenkel-Norbury model of isolated vortex ring, sometimes referred as the standard model, refers to the class of steady vortex rings having a linear distribution of vorticity in the core and parametrised by the mean core radius <math>\epsilon=\sqrt{A/\pi R^2}</math>, where <math>A</math> is the area of the vortex core and <math>R</math> is the radius of the ring. Approximate solutions were found for thin-core rings, i.e. <math>\epsilon\ll 1</math>,<ref name="fraenkel1970">{{cite journal |last1=Fraenkel |first1=L. E. |date=1970 |title=On steady vortex rings of small cross-section in an ideal fluid |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1970.0065 |journal=Proceedings of the Royal Society A |volume=316 |issue=1524 |pages=29–62 |doi=10.1098/rspa.1970.0065|bibcode=1970RSPSA.316...29F |s2cid=119895722 |url-access=subscription }}</ref><ref name="fraenkel1972">{{cite journal |last1=Fraenkel |first1=L. E. |date=1972 |title=Examples of steady vortex rings of small cross-section in an ideal fluid |url=https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/examples-of-steady-vortex-rings-of-small-crosssection-in-an-ideal-fluid/002BCAA9D14644C8B0232D99400B2AE0 |journal=Journal of Fluid Mechanics |volume=51 |issue=1 |pages=119–135 |doi=10.1017/S0022112072001107|bibcode=1972JFM....51..119F |s2cid=123465650 |url-access=subscription }}</ref> and thick Hill's-like vortex rings, i.e. <math>\epsilon\rightarrow\sqrt{2}</math>,<ref name="norbury1972">{{cite journal |last1=Norbury |first1=J. |date=1972 |title=A steady vortex ring close to Hill's spherical vortex |url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/steady-vortex-ring-close-to-hills-spherical-vortex/8F4871CDDBEE366E8AC9C308B8DC465B |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |volume=72 |issue=2 |pages=253–284 |doi=10.1017/S0305004100047083|bibcode=1972PCPS...72..253N |s2cid=120436906 |url-access=subscription }}</ref><ref name="norbury1973">{{cite journal |last1=Norbury |first1=J. |date=1973 |title=A family of steady vortex rings |url=https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/family-of-steady-vortex-rings/0BD7F8ACCA278BAD7B4C4B6E6977C355 |journal=Journal of Fluid Mechanics |volume=57 |issue=3 |pages=417–431 |doi=10.1017/S0022112073001266|bibcode=1973JFM....57..417N |s2cid=123479437 |url-access=subscription }}</ref> Hill's spherical vortex having a mean core radius of precisely <math>\epsilon=\sqrt{2}</math>. For mean core radii in between, one must rely on numerical methods. Norbury (1973)<ref name="norbury1973"/> found numerically the resulting steady vortex ring of given mean core radius, and this for a set of 14 mean core radii ranging from 0.1 to 1.35. The resulting streamlines defining the core of the ring were tabulated, as well as the translational speed. In addition, the circulation, the hydrodynamic impulse and the kinetic energy of such steady vortex rings were computed and presented in non-dimensional form.

===Instabilities===
A kind of azimuthal radiant-symmetric structure was observed by Maxworthy<ref>Maxworthy, T. J.  (1972) ''The structure and stability of vortex ring'', Fluid Mech. Vol. 51, p. 15</ref> when the vortex ring traveled around a critical velocity, which is between the turbulence and laminar states.  Later Huang and Chan<ref>Huang, J., Chan,  K.T. (2007)  ''Dual-Wavelike Instability in Vortex Rings'', Proc. 5th IASME/WSEAS  Int. Conf. Fluid Mech. & Aerodyn., Greece</ref> reported that if the initial state of the vortex ring is not perfectly circular, another kind of instability would occur. An elliptical vortex ring undergoes an oscillation in which it is first stretched in the vertical direction and squeezed in the horizontal direction, then passes through an intermediate state where it is circular, then is deformed in the opposite way (stretched in the horizontal direction and squeezed in the vertical) before reversing the process and returning to the original state.{{Citation needed|date=June 2021}}