Exact solutions to the principal variational inverse boundary value problem of aerohydrodynamics

Abstract

The problems discussed in this paper are related to a general question of aerohydrodynamics, which can be formulated in the following manner: what maximum lift can be attained by an isolated airfoil, and what should be the shape of the airfoil? The exact solution to the lift-optimization problem for an infinitely thin airfoil with a given length and limited curvature for a flow of an ideal incompressible fluid (IIF) is obtained in [1]. It is proved that the extremal airfoil shape is a circular segment. A review of methods and results related to the design of high-lift airfoils is available in [2]. A numerical approach to lift maximization for airfoils with sharp trailing edge and a specified contour perimeter under the condition of flow continuity is proposed in [3]. It is stated (without proof) that, for a smooth airfoil contour, the maximum lift is attained for a circle. A particular case of this is proved in [4]. Under additional conditions (e.g., the condition of viscous flow continuity on the airfoil contour, the allowance for flow compressibility, etc.), the optimized solutions significantly differ from a circle and the airfoil shape can be obtained only by numerical calculation (see, e.g., [5–7]). Nevertheless, the circle is an extremal analytical solution obtained under a minimum number of constraints stipulated by a mathematical flow model. Correspondingly, this solution yields the exact upper estimate of the maximum lift coefficient for the model based on an IIF. The problems investigated in [4–7] (see also [8]) relate to the class of variational inverse boundary value problems of aerohydrodynamics (IBVPA). Formulations and methods of solving these types of problems in the framework of the classical models of fluid mechanics and gas mechanics under isoperimetric constraints are discussed in [9]. There are also estimates of the