Power-series solution for the two-dimensional inviscid flow with a vortex and multiple cylinders

Abstract

The problem of a point vortex and N fixed cylinders in a two-dimensional inviscid fluid is studied and an analytical-numerical solution in the form of an infinite power series for the velocity field is obtained using complex analysis. The velocity distribution for the case of two cylinders is compared with the existing results of the problem of a vortex in an annular region which is conformally mapped onto the exterior of two cylinders. Limiting cases of N cylinders and the vortex, being far away from each other are studied. In these cases, "the dipole approximation" or "the point-island approximation" is derived, and its region of validity is established by numerical tests. The velocity distribution for a geometry of four cylinders placed at the vertices of a square and a vortex is presented. The problem of vortex motion with N cylinders addressed in the paper attracted attention recently owing to its importance in many applications. However, existing solutions using Abelian function theory are sophisticated and the theory is not one of the standard techniques used by applied mathematicians and engineers. Moreover, in the N ≥ 3 cylinder problem, the infinite product involved in the presentation of the Schottky-Klein prime function must also be truncated. So, the approach used in the paper is simple and an alternative to existing methods. This is the main motivation for this study.

Acoustic multiple scattering using recursive algorithms

“…They applied the algorithm to generate a grid for a finite difference scheme, and to numerically solve acoustic MS problems and estimate the pressure field over geometrically complex configurations. Pashaev and Yilmaz [20] studied the hydrodynamic interaction between an arbitrary number of cylinders and vortices. The problem is formulated using a Laurent series expansion with unknown coefficients which are found using a standard LU decomposition and the convergence properties of the infinite system was investigated.…”

Motion of vortices outside a cylinder

“…The basic idea behind these studies is that incoming waves can be decomposed into modes and diffracted waves from cylinders can be related to these modes and then the interaction takes place when we relate the coordinate systems at the center of each cylinder by using addition theorems. A similar idea was used by Pashaev and Yilmaz 7 to find the interaction between cylinders and a vortex in the twodimensional plane. There are other methods such as Abelian function theory and conformal mapping techniques used in solving vortex body interaction problems ͑see Refs.…”

µ-diff: A matlab toolbox for multiple scattering problems by disks

“…Even if disks can be considered as simple geometries, a reliable and highly accurate solution is required for wave propagation problems (acoustics, electromagnetics, optics, nanophotonics, elasticity) that involve many circular scatterers, modeling structured or disordered media, most particularly when k and M are large (see e.g. [9,15,19,20,21,22,24,33,34,35,36,39,40,42,45,49,50,52]). Let us note that all the developments in this paper directly apply to 2D TM/TE electromagnetic scattering waves [37] even if our presentation is more related to acoustics.…”