Lamb–Chaplygin dipole

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]

The flow structure of the Lamb-Chaplygin dipole

The model

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

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

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

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

where ${\displaystyle J_{0}{\text{ and }}J_{1}}$ are the zeroth and first Bessel functions of the first kind, respectively. Further, the value of ${\displaystyle k}$ is such that ${\displaystyle kR=3.83...}$, 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]

References

1. 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. 272: 157–182. doi:10.1017/S0022112094004428. ISSN 1469-7645.
2. Orlandi, Paolo (August 1990). "Vortex dipole rebound from a wall". Physics of Fluids A: Fluid Dynamics. 2 (8): 1429–1436. doi:10.1063/1.857591. ISSN 0899-8213.
3. 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. doi:10.1070/rd2004v009n04abeh000293. ISSN 1560-3547.
4. 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. doi:10.1063/1.4881375. ISSN 1070-6631.