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] This dipole is the two-dimensional analogue of Hill's spherical vortex.

A two-dimensional (2D), solenoidal vector field

may be described by a scalar stream function

is the right-handed unit vector perpendicular to the 2D plane.

By definition, the stream function is related to the vorticity

The Lamb–Chaplygin model follows from demanding the following characteristics: [citation needed] The solution

), 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

are the zeroth and first Bessel functions of the first kind, respectively.

, the first non-trivial zero of the first Bessel function of the first kind.

[citation needed] 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.