The pinch-off process in a starting buoyant plume
T. S. Pottebaum, M. Gharib
Abstract The vortex ring formation process of a starting
buoyant plume was studied experimentally using digital
particle image thermometry and velocimetry (DPITV). The
vortex ring was observed to pinch-off, or become discon-
nected, from the trailing plume. Pinch-off occurred at non-
dimensional times, or formation numbers, between 4.4 and
4.9. The observed pinch-off process is consistent with an
explanation based upon the Kelvin–Benjamin variational
principle. This is analogous to the pinch-off of a vortex ring
generated using a piston–cylinder apparatus, suggesting
that pinch-off is a general component of the vortex ring
formation process for various generation mechanisms.
Starting buoyant plume
The starting buoyant plume is a basic convection phe-
nomenon that occurs in a wide variety of circumstances.
The source of energy for a starting plume is the distribution
of mass in a gravity ﬁeld—available gravitational potential
energy is converted into kinetic energy as mass is redis-
tributed to minimize the total potential energy of the sys-
tem. Starting buoyant plumes are central to many
environmental ﬂows, such as cloud formation, discharge of
volcanic gases and the release of pollutants from industrial
and military sites. They are also the building blocks of more
complicated processes, like heat transfer from horizontal
boundaries and the Rayleigh–Taylor instability.
Turner (1962) introduced the term ‘‘starting plume’’ to
distinguish this type of ﬂow from both steady plumes and
buoyant vortex rings (thermals). Steady plumes are the
convection driven by a continuous source of buoyancy.
Thermals, in contrast, are the result of the sudden release
or production of a ﬁnite amount of buoyant ﬂuid. As a
starting plume develops, a cap forms that contains a vortex
ring similar to a thermal. Behind the cap, there is a stem or
trailing plume that resembles a steady plume. Turner
(1962) modeled the starting buoyant plume by assuming
that the cap moves with some ﬁxed fraction of the steady
plume velocity at the same height, and then calculated the
ﬂux of buoyancy and momentum from the stem to the cap
using the solution for a steady plume. The speciﬁc models
of a steady plume and of a buoyant vortex ring were taken
from Morton et al. (1956) and Turner (1957). This starting
plume model captures the essential difference between a
thermal and the vortex ring that forms as the cap of a
starting plume—the ﬂuid entrained by the cap from the
rear is ﬂuid from the stem, while a thermal entrains
ambient ﬂuid. Additional works on starting buoyant
plumes are reviewed by Fay (1973) and List (1982).
More recently, detailed experiments have been con-
ducted to measure the properties of the starting plume
directly. Tanny and Shlien (1986) measured the velocity
ﬁeld in a starting plume by tracing particle paths in long
exposure photographs. They found that vorticity in the cap
was diffused, with no distinct core. Chay and Shlien (1986)
measured the scalar ﬁeld (temperature or concentration)
in starting buoyant plumes using interferograms, and they
found that the scalar distribution depends on the Prandtl
number. Lundgren et al. (1992) conducted experiments
and computations of a downburst, and presented scaling
arguments to extend their results to atmospheric ﬂows.
Their experiments indicate that the cap velocity is constant
over a large period of time. From their computations, the
authors were able to present time resolved position and
circulation data for the cap of the downburst. Moses et al.
(1993) studied the scaling laws applicable to the cap.
Though they used encapsulated thermochromic liquid
crystal particles to visualize the starting plume, they did
not extract quantitative temperature or velocity data.
Vortex ring pinch-off
Gharib et al. (1998) performed a series of experiments with
vortex rings produced using a piston–cylinder apparatus.
They found that there exists a maximum circulation that
the vortex ring can attain, and any additional circulation
produced by the apparatus ends up in a trailing jet. The
critical parameter for this pinch-off process was identiﬁed
as the ‘‘formation number,’’ a non-dimensional time that
is equivalent to the stroke ratio, L/D, where L is the dis-
tance traveled by the piston and D is the cylinder diameter.
Gharib et al. (1998) showed that for formation numbers
less than about 4, all of the circulation produced by the
Experiments in Fluids 37 (2004) 87–94
Received: 18 October 2002 / Accepted: 25 December 2003
Published online: 25 March 2004
Ó Springer-Verlag 2004
T. S. Pottebaum (&), M. Gharib
Graduate Aeronautical Laboratories,
California Institute of Technology,
M/C 205–45, Pasadena, CA 91125, USA