Harmonic streamline of a thin bounded wing by subsonic flow of compressible gas

Abstract

A mathematical model of the perturbed velocity field caused by the flow of a compressible gas incident at a constant subsonic velocity on a thin weakly bent rigid bounded wing of arbitrary geometry in the plan is considered. The dependence of all functions on time is assumed to be periodic. The solution of the boundary value problem for the differential equation of elliptic type in three-dimensional space is represented by the double layer potential. The potential density corresponds to the pressure jump on the wing and can be found from the boundary condition for a given normal component of the disturbance velocity field on the wing. Satisfying this condition leads to an integral equation of the second kind in the wing domain.

The main purpose of this work is to establish the limit expressions for the normal derivative of the dipole potential. Separately, the integrals that make up this derivative have no limits, but the limit to their sum exists. The integral form for the limit expression that relates the pressure jump at the wing point to the integral mean values of this jump over the expanding circles of the perturbed velocity field propagation is found.

This work describes a mathematical model of the wing motion over a solid surface. Its solution is presented in an integral form. Taking into account the solid screen in the differential problem leads to additional terms in the integral equation, but these integrals are not relevant.

The integral equation of the second kind obtained in this work allows the application of iterative searching method of approximate solution.