Generalized functions on Hilbert spaces, singular integral equations, and problems of aerodynamics and electrodynamics

Abstract

External flow suction devices are used in aerodynamics to increase the lift of an airfoil in an ideal incompressible flow [1]. When solving this problem, the airfoil is modeled by a vortical layer [2], and the external flow suction device is modeled by a sink [3]. The desired velocity field around the profile should have the following property. In a neighborhood of the sink, the velocity field should have a sink-type singularity on the external side of the airfoil (where the sink lies), and it should be smooth on the opposite side. Naturally, if the airfoil satisfies the no-flow condition on the sink side, the sink point itself is excluded from this condition. The airfoil flow problem is thereby reduced to the solution of a singular integral equation on the profile contour with right-hand side undefined at the sink point. It follows from the analysis of tangential components of the velocity field in a neighborhood of the sink on the airfoil side where the velocity field is smooth that the solution of the resulting singular integral equation should be sought in the class of functions that have a singularity of the type of 1/x at the sink point [1]. In the numerical solution of this singular integral equation by the discrete vortex method, grid points are chosen so as to ensure that the sink point is one of them, and when replacing the singular integral equation by a system of linear algebraic equations, the equation corresponding to the sink point is omitted [1]. The omitted equation is replaced by another equation, which is derived from some physical considerations. This proves to be inconvenient, since, when using several external flow suction devices, one always has to decide how to fill in the missing equations. Hence it was suggested in [4] to satisfy the no-flow condition not on the side of the airfoil, where the sink lies, but on the opposite side, where the velocity field is smooth. This approach results in the appearance of a delta function supported at the sink point on the right-hand side of the corresponding singular integral equation. Now the singular integral equation should be treated as a pseudodifferential equation in the class of distributions. A version of such interpretation for a singular integral equation with a Hilbert kernel in the class of periodic distributions was presented in [5]. In this case, periodic distributions were treated as a subset of distributions on the entire real line. But this is inconvenient for singular integral equations on an interval in the class of distributions. However, it became simpler to use the discrete vortex method for the numerical solution of singular integral equations for the case in which the righthand side contains a delta function. The method acquired the same classical form [1] as for singular integral equations in the class of absolutely integrable functions, the delta function being replaced by the corresponding step function [6]. Note also that the solution of the problem on the computation of the input resistance of a thin wire antenna [7] powered by a current source leads to the analysis of a hypersingular integral equation on an interval, where the right-hand side contains a function with a singularity of the type of 1/x inside the solution domain. The solution of the corresponding characteristic equation was performed with the use of a spectral relation for a hypersingular integral operator on an interval, which provided a solution with a jump discontinuity at the point of the singularity of the right-hand side. In general, this solution should be considered in the class of distributions. This extension of the classical interpretation of singular and hypersingular integral equations and the corresponding