A calculus for flows in periodic domains
Abstract
We present a constructive procedure for the calculation of 2D 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 SchottkyKlein 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.00859
 Bibcode:
 2020arXiv200100859B
 Keywords:

 Physics  Fluid Dynamics;
 Mathematics  Complex Variables
 EPrint:
 29 pages