Power-series solution for the two-dimensional inviscid flow with a vortex and multiple cylinders |
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Authors: | Oktay K Pashaev Oguz Yilmaz |
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Affiliation: | (1) Lab. de Chimie quantique et Photophysique, Univ. Libre de Bruxelles, 1050 Bruxelles, Belgium |
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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. |
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