Periodic Schwarz–Christoffel mappings with multiple boundaries per period

Abstract

We present an extension to the theory of Schwarz–Christoffel (S–C) mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal (straight-line) boundaries per period. Taking the arrangements to be periodic in the x -direction in an ( x , y )-plane, three cases are considered; these differ in whether the period window extends off to infinity as y → ± ∞, or extends off to infinity in only one direction ( y → + ∞ or y → − ∞), or is bounded. The preimage domain is taken to be a multiply connected circular domain. The new S–C mapping formulae are shown to be expressible in terms of the Schottky–Klein prime function associated with the circular preimage domains. As usual for an S–C map, the formulae are explicit but depend on a finite set of accessory parameters. The solution of this parameter problem is discussed in detail, and illustrative examples are presented to highlight the essentially constructive nature of the results.

A calculus for flows in periodic domains

“…The first step in the procedure is to construct a conformal mapping from a circular domain to the physical target domain. A constructive formula for such mappings is available in [7]. The second step in the procedure is to construct the complex potential for the flow in the circular domain.…”

“…Further work has been done to solve the parameter problem in multiply connected domains [53]. Recent work [7] has further extended the original S-C mapping to permit target domains that are periodic. Similarly to other work by Crowdy [15,17], the mapping formula is phrased in terms of the Schottky-Klein prime function.…”

“…Consequently, the formula is valid for any number of objects per period window. The mapping formula is given by (6.3) in [7] as…”

“…In tandem with the Kutta condition, these vortices have a significant effect on the outlet angle of the flow, indicating that vorticity can have a significant effect on lift on cascades. More complicated blade shapes also can be modelled using the periodic Schwarz-Christoffel formula (2.3.5) as illustrated in [7].…”

Lightning Solvers for Potential Flows

“…At present, this problem cannot be solved with conformal maps, because the form of the required (polycircular) maps has not yet been found. Indeed, the general form of periodic polygonal maps has only been found recently [55]. Accordingly, the lightning approach provides a new method for tackling flows in periodic domains that were previously inaccessible.…”