Lamb–Chaplygin dipole

File:Lamb-Chaplygin_dipole.png

The Lamb–Chaplygin dipole model is a mathematical description for a particular inviscid and steady dipolar vortex flow. It is a non-trivial solution to the two-dimensional Euler equations. The model is named after Horace Lamb and Sergey Alexeyevich Chaplygin, who independently discovered this flow structure.{{Cite journal|last1=Meleshko|first1=V. V.|last2=Heijst|first2=G. J. F. van|date=August 1994|title=On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid|journal=Journal of Fluid Mechanics|language=en|volume=272|pages=157–182|doi=10.1017/S0022112094004428|bibcode=1994JFM...272..157M |s2cid=123008925 |issn=1469-7645}} This dipole is the two-dimensional analogue of Hill's spherical vortex.

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The model

A two-dimensional (2D), solenoidal vector field $\mathbf{u}$ may be described by a scalar stream function $\psi$ , via $\mathbf{u} = -\mathbf{e_z} \times \mathbf{\nabla} \psi$ , where $\mathbf{e_z}$ is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream function is related to the vorticity $\omega$ via a Poisson equation: $-\nabla^2\psi = \omega$ . The Lamb–Chaplygin model follows from demanding the following characteristics: {{citation needed|date=March 2020}}

The dipole has a circular atmosphere/separatrix with radius $R$ : $\psi\left(r = R\right) = 0$ .
The dipole propages through an otherwise irrotational fluid ( $\omega(r > R) = 0)$ at translation velocity $U$ .
The flow is steady in the co-moving frame of reference: $\omega (r < R) = f\left(\psi\right)$ .
Inside the atmosphere, there is a linear relation between the vorticity and the stream function $\omega = k^2 \psi$

The solution $\psi$ in cylindrical coordinates ( $r, \theta$ ), in the co-moving frame of reference reads:

$\begin{align}
\psi =
\begin{cases}
\frac{-2 U J_{1}(kr)}{kJ_{0}(kR)}\mathrm{sin}(\theta) , & \text{for } r < R, \\
U\left(\frac{R^2}{r}-r\right)\mathrm{sin}(\theta), & \text{for } r \geq R,
\end {cases}
\end{align}$

where $J_0 \text{ and } J_1$ are the zeroth and first Bessel functions of the first kind, respectively. Further, the value of $k$ is such that $kR = 3.8317...$ , the first non-trivial zero of the first Bessel function of the first kind.{{citation needed|date=March 2020}}

Usage and considerations

Since the seminal work of P. Orlandi,{{Cite journal|last=Orlandi|first=Paolo|date=August 1990|title=Vortex dipole rebound from a wall|journal=Physics of Fluids A: Fluid Dynamics|language=en|volume=2|issue=8|pages=1429–1436|doi=10.1063/1.857591|bibcode=1990PhFlA...2.1429O |issn=0899-8213}} the Lamb–Chaplygin vortex model has been a popular choice for numerical studies on vortex-environment interactions. The fact that it does not deform make it a prime candidate for consistent flow initialization. A less favorable property is that the second derivative of the flow field at the dipole's edge is not continuous.{{Cite journal|last1=Kizner|first1=Z.|last2=Khvoles|first2=R.|date=2004|title=Two variations on the theme of Lamb–Chaplygin: supersmooth dipole and rotating multipoles|journal=Regular and Chaotic Dynamics|volume=9|issue=4|pages=509|doi=10.1070/rd2004v009n04abeh000293|issn=1560-3547}} Further, it serves a framework for stability analysis on dipolar-vortex structures.{{Cite journal|last1=Brion|first1=V.|last2=Sipp|first2=D.|last3=Jacquin|first3=L.|date=2014-06-01|title=Linear dynamics of the Lamb-Chaplygin dipole in the two-dimensional limit|journal=Physics of Fluids|volume=26|issue=6|pages=064103|doi=10.1063/1.4881375|bibcode=2014PhFl...26f4103B |issn=1070-6631|url=https://hal.archives-ouvertes.fr/hal-01100934/file/DAFE14028.1418984393.pdf}}

References

Category:Fluid dynamics