I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2006, VOLUME 12, NUMBER 4, PAGES 65-77
Helmholtz--Kirchhoff method and boundary control of a plane flow
A. S. Demidov
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We consider the problem of the description of the eddy singularities
behind an obstruction flowed around and their
dependence (and hence, the dependence of flow functionals, e.g., the
resistance force) on parameters determining the
boundary of the obstacle
and/or flow characteristics on .
We propose a new approach to these problems for a flat
potential flow of an incompressible liquid, which is based on ideas of
the Helmholtz--Kirchhoff method and the Euler equation under the
assumption that the flow has a point vortices concentrated at the
required centers , where the
potential of the velocity
(, ) has
a singularity proportional to .
In the case of a -segment polygonal
obstacle and a (chosen in some way) number of point vortices taken
into account in calculations, the flow can be reconstructed by the
so-called characteristic values of the potential.
It occurs that, being the components of the required vector function
are connected by certain functional equations corresponding to
geometric properties of the obstacle, intensity of vortices, frequency
of their breakdown from the obstacle, etc.
These equations involve the Helmholtz--Kirchhoff function specified on the
Riemannian surface .
This surface and the boundary conditions for the function are parameterized
by the function and by a control
defined on .
As for the pressure , it is defined by the
Cauchy--Lagrange equation for the Euler equation.
Last modified: February 17, 2007