Treatment of Neumann boundaries in the complex variable boundary element method

摘要

For potential flow, the complex variable boundary element method (CVBEM) is formulated in terms of the velocity potential Φ and the stream function Ψ. In actual flow problems, Φ and partial deriv Φ/partial deriv n are given along Dirichlet and Neumann boundaries, respectively. In the CVBEM, the Neumann-type condition partial deriv Φ/partial deriv n is not directly handled, and, instead, Ψ is used to define Neumann boundaries. Owing to this discrepancy, numerical difficulties are raised along Neumann boundaries. The current study addresses two such difficulties: (1) multiple Neumann boundaries and (2) branch cuts across Neumann boundaries. The first problem is due to the fact that Ψ along multiple boundaries cannot be specified a priori, and the second problem is due to the discontinuous jump inherent in Ψ for sink/source singularities. To overcome these difficulties, a new formulation of the CVBEM to solve for the unknown Ψ values and a proper way of branch-cut placement are proposed, and these techniques arc verified against example problems.