Slender body expansions in potential theory along a finite straight line

Abstract

Consider a distribution of singularities in a potential field along a finite straight line such that the potential satisfies the Laplace equation. An example is a distribution of sources representing a ship or missile moving with forward velocity in a potential inviscid flow field. Such bodies are often truncated or bluff at the ends, and so the strength of the resulting distributions may not gradually tend to zero close to these ends and may instead be non-zero finite. A near-field expansion is obtained which accounts for this using the slender body theory integral splitting method. All terms in the expansion are obtained, and the coefficient of each term in the infinite sequence is given in terms of differentials of the distribution strength. Hence an exact separation of variables solution (separating the axial distance from the cross-sectional distances) is obtained for the potential. This is different from previous representations in that it represents a distribution over a finite length, and the resulting expansion is a simple single summation expression that is straightforward to apply. The resulting numerical scheme is discussed, in particular the evaluation close to the ends and also a comparison between the presented slender body theory and existing numerical methods.