Applications of degenerate kernels to potential flow across circular, elliptical cylinders and a thin airfoil

Abstract

In this paper, we employ the boundary integral equation (BIE) to analytically derive the solution of potential flow across a circular cylinder, an elliptical cylinder and a thin airfoil. It is found that the symmetric or antisymmetric potential flow problem can be solved by using the UT or the LM equation alone. Both analytical and numerical approaches were considered. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using polar and elliptical coordinates for a circular cylinder and an elliptical cylinder, respectively. The role of the dual BEM is also examined by showing the singular vectors of influence matrix. Besides, either the singular or the hypersingular BIE are respectively employed alone to solve the problem of symmetric and anti-symmetric flow field. Analytical derivation as well as the BEM is demonstrated. The orientation of the ellipse, as well as the angle of attack for the thin airfoil, is considered. The result shows that the degenerate kernel can be an alternative tool for solving some BVPs, although the complex variable is always employed. Once the degenerate kernel for the closed-form fundamental solution is available, the BIE is nothing more than the linear algebra and an analytical study is possible. The extension to 3-D object is promising once the degenerate kernel is available.