## Motion of a two-dimensional monopolar vortex in a bounded rectangular domain

J.H.G.M. van Geffen, V.V. Meleshko and G.J.F. van Heijst
Physics of Fluids 8, 2393-2399 (1996)

### I. Introduction

Two-dimensional (2D) vortex motion in bounded domains has been studied since quite a long time (Villat [1]; Müller [2]; see also Saffman [3]). Traditionally, the attention was concentrated on the motion of point vortices in an inviscid fluid with so-called "free-slip" walls: the normal component of the velocity is equal to zero at the wall, with no restriction on the tangential velocity. An elegant mathematical technique based on Green's function (Saffman [3]) permits to write a set of ordinary differential equations for the motion of point vortices inside any domain. These equations express in a rather general and concise manner a very extensive class of phenomena. One important conclusion is that a single point vortex, although immovable in an unbounded fluid at rest at infinity, placed in a bounded domain will move due to the velocity field induced by the system of its image vortices.

On the other hand, comparitively little has been done to clarify the influence of viscous effects on the motion of initially compact vorticity distributions in a bounded domain. This paper provides a comparative analysis of the motion of a point vortex and of a circular vortex with a non-singular initially axisymmetric vorticity distribution (henceforth referred to as "monopole") in an inviscid and viscous fluid, respectively, confined in a rectangular domain with free-slip walls. Such a rather simple and basic configuration offers a better understanding of the possibilities of both models.

In physical reality, however, free-slip walls are not present since there is always friction at the walls. Hence, one would want to apply a "no-slip" boundary condition: at the wall the velocity of the fluid equals zero. This condition implies generation of oppositely-signed vorticity near the wall, leading to a flow evolution different from the case of free-slip walls (see e.g. Orlandi [4]; Verzicco et al. [5]). A no-slip boundary condition cannot be applied to point vortices, but it can be applied in the method used for the computations with the distributed monopole, so that the effect of no-slip walls can be studied too.

In the next section the vortex models used in the numerical simulations are discussed. The results of these simulations for free-slip and no-slip domain boundaries are presented in Sect. III and IV, respectively. The paper ends with some general conclusions.

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