Numerical Study of Vortex Ring Evolution and Free Surface Interaction. 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 . 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.  and Quyuan & Chu  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  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. . 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. References  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.  M. Song, L.P. Bernal, and G. Tryggvason. Head-on collision of a large vortex ring with a free surface. 4:1457–1466, 1992.  Y. Quyuan and C.K. Chu. The nonlinear interaction of vortex rings with a free surface. Acta Meccanica Sinicia, 13:120–129, 1997.  M. Gharib and A. Weigand. Experimental study of vortex disconnection and connection at a free surface. Journal of Fluid Mechanics, 321:59–86, 1996. 1 2 z y 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. y/R0 x/R0 z/R0 Ring PSfrag replacements Hairpins y/R0 x/R0 2 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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