Solution of an inverse boundary value problem in aerohydrodynamics (Q2479651)

It is well-known that designing the wing section of an airfoil boat, even within the model of an ideal incompressible fluid (IIF), one encounters some mathematical difficulties. The latter are caused by the ill-posedness of the problems and the double connectedness of the flow domain. The inverse problem is solved in [\textit{M. I. Galyautdinov} and \textit{D. V. Maklakov}, Designing of wing sections, streamlined near a hard screen, Izv. Vyssh. Uchebn. Zaved. Aviatsion. Tekhn. 1994, No. 3, 3--7 (1994)] with the help of elliptic functions; one defines the attack angle during the solution process. In [\textit{A. N. Il'inskii, N. B. Il'inskii, D. V. Maklakov} and \textit{A. V. Potashev}, A method for aerodynamic airfoil design, Izv. Vyssh. Uchebn. Zaved. Aviatsion. Tekhn. 1995, No. 2, 54--62 (1995)] one introduced a fictive plane-parallel flow of an IIF under the screen. This makes the flow domain simply connected, but the function of the complex potential becomes piecewise-analytic, discontinuous at the screen. One finds it by a rather complicated iterative process, also determining the attack angle during the solution process. In the problem under consideration the lower surface of the wing section is assumed to be flat; its inclination to the screen is given (in fact, this angle is close to the attack angle). We propose a method which allows one to proceed from a biconnected flow domain to a simply connected, but two-sheeted one. This approach enables us to reduce the initial boundary value problem to the boundary value problem in the simply connected domain. Namely, we obtain the inverse boundary value problem (IBVP) for a symmetric wing section with the withdrawal channel near the leading edge and the blowing channel at the trailing edge of the airfoil in an unbounded flow.We have succeeded in writing a solution explicitly;we have coped with the solvability problem, introducing free parameters in the initial velocity distribution. We performed several numerical tests. We verified the results, solving the direct problem [\textit{D. V. Maklakov}, Nonlinear hydrodynamics problems for potential flows with unknown boundaries. (Russian) Moscow: Yanus-K (1997; Zbl 0889.76003), p. 159], and made certain of the high accuracy of the developed method. We adduce an example of the airfoil construction and make conclusions on the applicability of the proposed method.