Numerical Study of Vortex Ring Evolution and Free Surface Interaction by asafwewe


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									     Numerical Study of Vortex Ring Evolution and Free Surface
                           P.J. Archer, T.G. Thomas, and G.N. Coleman
                  School of Engineering Sciences, University of Southampton.
   Vortex rings are interesting because, apart from their ubiquitous nature, their growth,
instability and breakdown embodies a prototypical turbulent flow. Our previous numerical
investigations into the unbounded evolution of initially laminar vortex rings have shown
three characteristic phases in the vortex ring lifetime: laminar, transitional and turbulent.
A laminar ring, of radius R, is typified by a single toroidal vortex filament, of core thickness
δ and circulation Γ, and is susceptible to the Widnall instability [1]. The instability distorts
the inner core region into stationary azimuthal wave, however the outer core region (which
we call halo vorticity) moves out of phase of the inner core and forms discrete loops wrapped
around the wavy inner core (figure 1(a)). Neighbouring loops are of alternating sign vorticity
and their shedding, as a series of hairpin vorticies, marks the onset of turbulence (figure 1(b)).
For turbulent rings evolved in this way, the structure of the ring depends on the relative
thickness of the core region prior to transition. Thin core rings (δ/R ≈ 0.25) maintain a
coherent core region encircled by smaller scale vorticity filaments, whereas thick core rings
(δ/R > 0.35) become a tangle of interwoven vorticity filaments with no defined core region.
   We now consider the free-surface interaction of the vortex ring, focusing on the effect of
the different phases of the ring life cycle. Previous work on laminar rings by Song et al. [2]
and Quyuan & Chu [3] have shown intricate vortex dynamics at the free surface, including
reconnection of the vortex core to the free-surface. For the oblique case, Weigand and
Gharib [4] found multiple surface connections that could be associated with the organised
structures within the vortex core and the vortex filaments wrapped around it. We limit
the investigation to the normal interaction and consider laminar, transitional and turbulent
rings. The direct numerical simulations are initialised by extraction of a vorticity field from
the precursor simulations of the unbounded vortex ring, which is then embedded at a depth
of 4R0 below a deformable free surface.
   The results for the interaction of a laminar ring show good agreement with the experi-
ments of Song et al. [2]. As the ring approaches a depth of one radius the free surface above
the ring is locally deformed into a bulge. Nearing the surface the ring begins to interact with
its virtual image, altering the ring trajectory until the ring expands in the plane parallel
to the surface at a small depth of order δ. Correspondingly the surface bulge drops and a
surface depression is formed outboard of the ring. The interaction of the transitional ring
is modified by the presence of the vortical loops of halo vorticity. As the ring expands be-
neath the surface the loops disconnect and reconnect with their respective images above the
surface, forming a series of half rings and causing localised surface depressions. As the ring
continues to expand the wavy inner core also reconnects with the surface. The interaction
of a turbulent ring shows multiple reconnection zones as the swirling core vorticity filaments
impinge on the surface (figure 2).
   The research is joint funded by EPSRC and DSTL.

[1] S.E. Widnall and C-Y. Tsai. The instability of the thin vortex ring of constant vorticity. Phil. Trans. R.
    Soc. Lond., 287:273–305, 1977.
[2] M. Song, L.P. Bernal, and G. Tryggvason. Head-on collision of a large vortex ring with a free surface.
    4:1457–1466, 1992.
[3] Y. Quyuan and C.K. Chu. The nonlinear interaction of vortex rings with a free surface. Acta Meccanica
    Sinicia, 13:120–129, 1997.
[4] M. Gharib and A. Weigand. Experimental study of vortex disconnection and connection at a free surface.
    Journal of Fluid Mechanics, 321:59–86, 1996.



PSfrag replacements                               PSfrag replacements
                  Wavy inner core                        Wavy inner core                  x
                                                      Halo vorticity
                                                            Halo vorticity
                                            (a)                                                    (b)

                         Figure 1. Transitional vortex structure visualised by isosurfaces of the second
                         invariant of the velocity gradient tensor II: (a) showing the development of vortical
                         loops wrapped around the wavy inner core region; (b) at a slightly later time the
                         outer loops are detrained into the wake as a string of hairpin vorticies. Thin core
                         ring, initial δ/R = 0.2 and Re = 7500.



 PSfrag replacements


                                         y/R0                                                 x/R0

                         Figure 2. Turbulent ring surface interaction at time t = 31.2Γ0 /R0 . Vorticity
                         structure below the surface is visualised by II = -0.005 isosurface. The free sur-
                         face distortion pattern is shown in the insert with light shading corresponding to
                         regions of elevation and dark to depression.

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