Nonuniform Steady Flow of an Ideal Fluid past Airfoils. Part I. Some Exact Solutions for Two-Dimensional Nonuniform Flow of Exponential Type

摘要

This paper is the first of a series in which the theory of wings, well established for uniform flow, shall be generalized to nonuniform flow. The main difficulty in nonuniform flow is that the Euler equations cannot be linearized exactly. For two dimensional flow there are only two exceptions: linear shear flow, which can be reduced to the Laplace equation and, therefore, does not lead to principally new problems, and a flow, called in this paper exponential. For this flow exact solutions by expansions into modified Bessel functions or Mathieu functions are given if the profile is a circle, an ellipse or a flat plane airfoil. For the latter the analogue of the so-called singularity method based on the concept of a vortex sheet is also developed. The solution is determined by a singular integral equation, for which a numerical method is outlined. The numerical results are compared with those of uniform and of linear shear flow. (Author)