Helmholtz--Kirchhoff method and boundary control of a~plane flow

A. S. Demidov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 65-77

Résumé

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 $\partial S$ of the obstacle and/or flow characteristics on $\partial S$. 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 a point vortices concentrated at the required centers $z_k^*$, 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 $\mathrm{arg}(z-z_k^*)$. 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 $$ \sigma\colon t\mapsto(\sigma_1(t),\ldots,\sigma_M(t))\in\mathbb R^M, \ \ \text{where}\ \ M =M(K,L), $$ 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 $Q=Q(\sigma)\ni w$. This surface and the boundary conditions for the function $\ln(dz/dw)$ are parameterized by the function $\sigma$ and by a control defined on $\partial S$. As for the pressure $p$, it is defined by the Cauchy–Lagrange equation for the Euler equation.