Visualization of Potential Flow: Dealing with the Branch Cuts

Abstract

The theory of potential flow is important in several application fields, in particular: in aero dynamics, fluid dynamics, electrostatics, porous media flow. In the classical approach flow is described by a potential function of the complex plane into the complex plane. Using potential function contouring the visualizing of streamlines in flow nets can reveal important characteristics of the flow in question, for which alternative methods fail. However, branch cuts of important potential functions pose a problem for the visual representation. Here methods are outlined how to deal with the problem: (1) to be aware that equalities, common in real number algebra, may not hold in the complex number space; (2) to partition the complex plane into sub-domains on which the functions are evaluated sequentially. The description of the methods uses superpositions of complex logarithms as examples, but the ideas can be adopted for the visualisation to other potentials as well.