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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. The model is named after Horace Lamb and
Sergey Alexeyevich Chaplygin Sergey Alexeyevich Chaplygin (russian: Серге́й Алексе́евич Чаплы́гин; 5 April 1869 – 8 October 1942) was a Russian and USSR, Soviet physicist, mathematician, and mechanical engineer. He is known for ...
, who independently discovered this flow structure. This dipole is the two-dimensional analogue of
Hill's spherical vortex Hill's spherical vortex is an exact solution of the Euler equations that is commonly used to model a vortex ring. The solution is also used to model the velocity distribution inside a spherical drop of one fluid moving at a constant velocity throu ...
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The model

A two-dimensional (2D), solenoidal vector field \mathbf may be described by a scalar
stream function The stream function is defined for incompressible ( divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. ...
\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 \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 t ...
: -\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 irrorational 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 (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 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