A calculus for flows in periodic domains

TL;DR

This paper introduces a constructive method for calculating exact 2-D potential flows in various periodic domains using conformal mapping and complex potentials, applicable to multiple obstacles and flow phenomena.

Contribution

It provides a unified, exact approach for potential flows in periodic domains with multiple boundaries, utilizing conformal maps and Schottky--Klein functions, without asymptotic approximations.

Findings

Solutions valid for arbitrary obstacles per period

Applicable to diverse flow phenomena including singularities and moving boundaries

Exact solutions without asymptotic approximations

Abstract

We present a constructive procedure for the calculation of 2-D potential flows in periodic domains with multiple boundaries per period window. The solution requires two steps: (i) a conformal mapping from a canonical circular domain to the physical target domain, and (ii) the construction of the complex potential inside the circular domain. All singly periodic domains may be classified into three distinct types: unbounded in two directions, unbounded in one direction, and bounded. In each case, we relate the target periodic domain to a canonical circular domain via conformal mapping and present the functional form of prototypical conformal maps for each type of target domain. We then present solutions for a range of potential flow phenomena including flow singularities, moving boundaries, uniform flows, straining flows and circulatory flows. By phrasing the solutions in terms of the transcendental Schottky--Klein prime function, the ensuing solutions are valid for an arbitrary number of obstacles per period window. Moreover, our solutions are exact and do not require any asymptotic approximations.