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periodic forcing. To this end the flow is assumed to be the sum of two point vortex and
a linear sink whose intensity is time dependent. They found that the dipole escapes
or returns to the channel outlet depending on the value of the Strouhal number. With
respect to experimental works, the properties and the evolution of the dipole have
been investigated in two cases: (a) a dipole produced by an impulsively jet (Sous et al.
2004
; Lacaze et al.
2010
) and (b) a dipole produced under periodic forcing (Nicolau
del Roure et al.
2009
). Finally, a numerical study of a periodic forced flow has been
performed by López-Sánchez and Ruiz-Chavarría (
2013
). They found that lifetime
of vortices extends over more than a driving period. In this case interaction between
two dipoles can occur, leading under certain circumstances to vortex coalescence.
In shallow-water, apart from the two counter-rotating vortices, a third vortex has
been reported, which is perpendicular to the dipole (Albagnac
2010
). In this respect,
Lacaze et al. (
2010
) have experimentally studied the properties of the spanwise vortex
produced by an impulsively jet. They found that all three vortices have comparable
intensities. On the other hand, Duran-Matute et al. (
2010
) made a numerical simula-
tion to study this structure in a thin horizontal layer. They assumed that initially the
velocity field is like a Lamb-Chaplygin vortex in the horizontal plane and in the verti-
cal plane the velocity field follows a Poiseuille profile. They found that, under certain
conditions, the 2D approach is no longer sufficient to describe this flow because 3D
effects are present. The three dimensional nature of the flow depends on a single
parameter
K
2
Re
, where
is the aspect ratio (the fluid depth to the size R of the
dipole) and Re is the Reynolds number. If
K
=
ʴ
ʴ
6 the flow is dominated by the vis-
cosity, so the vertical motion can be neglected. In the range 6
<
<
<
15, the dipole
properties are modified by the vertical motion and a spanwise vortex appears in front
of the dipole. Finally, for
K
K
15 the intensity of the spanwise vortex becomes com-
parable with those of the counter-rotating vortices. In the present work we make a
study of the spanwise vortex under periodic forcing. For comparison we also present
some results for a impulsively forcing. For this purpose we solve numerically the
Navier-Stokes and continuity equations with a finite volume method, using the free
software OpenFOAM. The originality lies in the fact that the flow rate is periodic.
The study of this spanwise vortex is important in coastal systems (where periodic
forcing is induced by tides) because it influences the lift, transport and deposition of
particles located at the bottom or inside the fluid.
This paper is organized as follow. Section
2
is devoted to describe the system
under study and the numerical method. In Sect.
3
we present some numerical results
of both impulsively and periodic forcing. Section
4
deals with the convergence of
the numerical code and in Sect.
5
the conclusions are drawn.
>
2 Description of the System and Numerical Simulation
The flow we study occurs in a channel and an open domain. In order to use Open-
FOAM the overall domain is decomposed in four parallelepiped, as shown in Fig.
1
.
The equations to be solved are the Navier-Stokes (
1
) and the continuity (
2
) equa-
tions for an incompressible flow. In dimensionless form, these equation are:
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