Numerical Construction of Potential Flows in Multiply Connected Channel Domains

Abstract

We consider inviscid and incompressible flows in two-dimensional multiply connected domains between two infinitely long parallel straight lines, which are simple models describing channel flows such as rivers, streams and canals. In the present paper, we propose two computational methods to construct the complex potentials for a uniform flow, a point vortex, a source and a sink in the multiply connected channel domains. First, we give analytic formulae of the potential flows in a special channel domain with circular holes, which are described in terms of the Schottky-Klein prime function as in Crowdy and Marshall [5]. Second, we make use of the numerical conformal mapping techniques by Amano [1] and Ogata [11] to approximate the complex potentials in the channel domains that contain boundaries of arbitrary shapes. We also confirm that the numerical techniques can be applied to construct the potential flows in a real river. The two methods provide us with analytic representations of the complex potentials, which enable us to compute the velocity field in the channel domains and the force on the boundaries quickly and accurately.