Orthogonal mapping

A technique of orthogonal mapping is proposed for constructing boundary-fitted orthogonal curvilinear coordinate systems in 2-D. The mapping is defined by the covariant Laplace equation, and constraints on the components of the metric tensor of the curvilinear coordinates are used to achieve orthogonality and to control the spacing of coordinate lines. Two different methods of implementing the mapping are presented. The first, termed the strong constraint method, is intended primarily for problems in which the boundary shape is not known in advance, but is to be determined as a part of the solution (e.g., free boundary problems in fluid mechanics). The second, termed the weak constraint method, is designed for the construction of an orthogonal mapping with a prescribed boundary correspondence, i.e., the production of boundary-fitted orthogonal coordinates for a domain of given shape with a prescribed distribution of coordinate nodes along the boundary. The method is illustrated by numerical examples, and it is shown that the problem of mapping infinite domains can be treated by mapping the infinite domain onto a finite one using a simple conformal transformation and then applying the orthogonal mapping technique developed here to the finite domain. The possibility of obtaining analytical solutions for the mapping functions is discussed. The Appendices contain connection (Christoffel) coefficients which provide a convenient means for deriving equations of a physical problem for the constructed coordinates in terms of physical components, using a slight extension of Cartesian tensor notation.