## 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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