Lamb–Chaplygin dipole explained
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.[1] This dipole is the two-dimensional analogue of Hill's spherical vortex.__TOC__
The model
may be described by a scalar
stream function
, via
, where
is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream function is related to the
vorticity
via a
Poisson equation:
. The Lamb–Chaplygin model follows from demanding the following characteristics:
- The dipole has a circular atmosphere/separatrix with radius
:
.
- The dipole propages through an otherwise irrorational fluid (
at translation velocity
.
- 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
The solution
in
cylindrical coordinates (
), in the co-moving frame of reference reads:
\begin{align}
\psi=
\begin{cases}
sin(\theta),&forr<R,\
U\left(
-r\right)sin(\theta),&forr\geqR,\end{cases}
\end{align}
where
are the zeroth and first
Bessel functions of the first kind, respectively. Further, the value of
is such that
, the first non-trivial zero of the first Bessel function of the first kind.
Usage and considerations
Since the seminal work of P. Orlandi,[2] 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.[3] Further, it serves a framework for stability analysis on dipolar-vortex structures.[4]
Notes and References
- Meleshko. V. V.. Heijst. G. J. F. van. August 1994. On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid. Journal of Fluid Mechanics. en. 272. 157–182. 10.1017/S0022112094004428. 1994JFM...272..157M . 123008925 . 1469-7645.
- Orlandi. Paolo. August 1990. Vortex dipole rebound from a wall. Physics of Fluids A: Fluid Dynamics. en. 2. 8. 1429–1436. 10.1063/1.857591. 1990PhFlA...2.1429O . 0899-8213.
- Kizner. Z.. Khvoles. R.. 2004. Two variations on the theme of Lamb–Chaplygin: supersmooth dipole and rotating multipoles. Regular and Chaotic Dynamics. 9. 4. 509. 10.1070/rd2004v009n04abeh000293. 1560-3547.
- Brion. V.. Sipp. D.. Jacquin. L.. 2014-06-01. Linear dynamics of the Lamb-Chaplygin dipole in the two-dimensional limit. Physics of Fluids. 26. 6. 064103. 10.1063/1.4881375. 2014PhFl...26f4103B . 1070-6631.
