Lamb–Chaplygin Dipole
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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 In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
. The model is named after
Horace Lamb Sir Horace Lamb (27 November 1849 – 4 December 1934R. B. Potts,, '' Australian Dictionary of Biography'', Volume 5, MUP, 1974, pp 54–55. Retrieved 5 Sep 2009) was a British applied mathematician and author of several influential texts on ...
and Sergey Alexeyevich Chaplygin, who independently discovered this flow structure. This dipole is the two-dimensional analogue of Hill's spherical vortex. __TOC__


The model

A two-dimensional (2D),
solenoidal vector field In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf ...
\mathbf may be described by a scalar
stream function In fluid dynamics, two types of stream function (or streamfunction) are defined: * The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, is defined for incompressible flow, incompressible (divergence-free ...
\psi, via \mathbf = -\mathbf \times \mathbf \psi, where \mathbf is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream function is related to the
vorticity In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point an ...
\omega via a
Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
: -\nabla^2\psi = \omega. The Lamb–Chaplygin model follows from demanding the following characteristics: * 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 A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
(r, \theta), in the co-moving frame of reference reads: \begin \psi = \begin \frac\mathrm(\theta) , & \text r < R, \\ U\left(\frac-r\right)\mathrm(\theta), & \text r \geq R, \end \end where J_0 \text J_1 are the zeroth and first
Bessel functions Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
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.


Usage and considerations

Since the seminal work of P. Orlandi, 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. Further, it serves a framework for stability analysis on dipolar-vortex structures.


References

{{DEFAULTSORT:Lamb-Chaplygin dipole Fluid dynamics