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]
![](../I/m/Lamb-Chaplygin_dipole.png)
The model
A two-dimensional (2D), solenoidal vector field 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: .
- 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:
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]
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. 272: 157–182. doi:10.1017/S0022112094004428. ISSN 1469-7645.
- 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.
- 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.
- 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.