Helmholtz-Kirchhoff method and boundary control of a plane flow

Abstract

We consider the problem of the description of the eddy singularities behind an obstruction S and their dependence (and hence, the dependence of flow functionals, e.g., the resistance force) on the 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 \(d\vec V/dt = \nabla p\) under the assumption that the flow has point vortices concentrated at the required centers zk*, where the potential u of the velocity \(\vec V = \overline {dw} /dz(w = u + iv \in \mathbb{C},z = x + iy)\) 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 the 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 $$\sigma :t \mapsto (\sigma _1 (t), \ldots ,\sigma _M (t)) \in {\mathbb{R}}^M , where M = M(K,L),$$ 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(σ) ∋ w. This surface and the boundary conditions for the function ln(dz/dw) are parametrized by the function σ and by a control defined on ∂S. As for the pressure p, it is defined by the Cauchy-Lagrange equation for the Euler equation.