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 S flowed around and their dependence (and hence, the dependence of flow functionals, e.g., the resistance force) on parameters determining the boundary S of the obstacle and/or flow characteristics on S. 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 dV/dt = Ñp under the assumption that the flow has a point vortices concentrated at the required centers zk*, where the potential u of the velocity V = \overline{dw}/dz (w = u + iv Î C, z = x + iy) has a singularity proportional to arg (z - zk*). In the case of a K-segment polygonal obstacle and a (chosen in some way) number L 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 s: t \mapsto (s1(t), ¼, sM(t)) Î RM, where M = M(K,L), they 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 ln (dz/dw) specified on the L-fold Riemannian surface Q = Q(s) ' w. This surface and the boundary conditions for the function ln (dz/dw) are parameterized by the function s and by a control defined on S. As for the pressure p, it is defined by the Cauchy--Lagrange equation for the Euler equation.

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Last modified: February 17, 2007