A study has been performed of the interaction of
periodic vortex rings with a central
columnar vortex, both for the case of identical vortex rings and the case
of rings of
alternating sign. Numerical calculations, both based on an adaptation of
the
Lundgren–Ashurst (1989) model for the columnar vortex dynamics and
by numerical
solution of the axisymmetric Navier–Stokes and Euler equations
in the vorticity–velocity formulation using a viscous
vorticity collocation method, are used to
investigate the response of the columnar vortex to the ring-induced
velocity field. In all
cases, waves of variable core radius are observed to build up on the columnar
vortex
core due to the periodic axial straining and compression exerted by the
vortex rings.
For sufficiently weak vortex rings, the forcing by the rings
serves primarily to set an
initial value for the axial velocity, after which the columnar vortex waves
oscillate
approximately as free standing waves. For the case of identical rings,
the columnar
vortex waves exhibit a slow upstream propagation due to the nonlinear forcing.
The
cores of the vortex rings can also become unstable due to the straining
flow induced
by the other vortex rings when the ring spacing is sufficiently small.
This instability causes the ring vorticity to spread out into a sheath
surrounding the columnar vortex.
For the case of rings of alternating sign, the wave in core
radius of the columnar vortex
becomes progressively narrower with time as rings of opposite sign approach
each
other. Strong vortex rings cause the waves on the columnar vortex to grow
until they
form a sharp cusp at the crest, after which an abrupt ejection of vorticity
from the
columnar vortex is observed. For inviscid flow with
identical rings, the ejected vorticity
forms a thin spike, which wraps around the rings. The thickness
of this spike increases
in a viscous flow as the Reynolds number is decreased. Cases have also
been observed,
for identical rings, where a critical point forms on the
columnar vortex core due to the
ring-induced flow, at which the propagation velocity of upstream waves
is exactly
balanced by the axial flow within the vortex core when
measured in a frame translating
with the vortex rings. The occurrence of this critical
point leads to trapping of wave
energy downstream of the critical point, which results in large-amplitude
wave growth
in both the direct and model simulations. In the case of
rings of alternating sign, the
ejected vorticity from the columnar vortex is entrained and carried
off by pairs of rings
of opposite sign, which move toward each other and radially outward under
their self- and mutually induced velocity fields, respectively.
The influence of turbulence on a columnar vortex with axial flow
