Abstract
Axisymmetric Green's functions and the associated stream functions, which satisfy the boundary condition that the wall of the tube is a stream surface, are presented for various singularities. These include a source on the axis an axially-oriented doublet, a source ring, a source disk, and a vortex ring. Results are expressed as integrals of the modified Bessel functions. These can be applied to formulate Fredholm integral equations of either the first or second kind for determining the axisymmetric, irrotational flow about bodies of revolution in a tube. In the present work, only those of the first kind are treated, including integral equations for axial source and doublet distributions, and a vortex sheet on the surface of the body. Three different methods for solving each of these three integral equations are examined: the method of piecewise-constant singularity elements of von Karman, Kaplan's method of expanding the unknown distribtuion as a series of Legendre polynomials, and solutions by a technique of eliminating peaks in the kernel (representing integrals by means of a quadrature formula) and solving the resulting set of linear equations by means of a suitable iteration formula. Numerical results for three of the methods, applied to a spheroid, are presented. The resulting added masses and a comparison with predictions from slenderbody theory are given in Appendixes.
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