
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 ...
.
__TOC__
The model
A two-dimensional (2D),
solenoidal vector field 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. ...
, 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
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 ...
:
. 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, 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