Editing Vortex ring (section)

===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.