Density-driven
environmental flows
Personnel
James Rottman (Adjunct professor, UCSD)
Ryan Lowe (former undergraduate at UCSD,
now graduate student at Stanford)
Eckart Meiburg (UCSB)
Jake Hacker (Arup)
Stuart Dalziel
(DAMTP)
Jonathan Shin (former PhD student, DAMTP)
1 Gravity
currents
Gravity currents occur when there are horizontal variations
in density in a fluid under the action of a gravitational field. A simple
example that can readily be experienced is the gravity current that flows into
a warm house through a doorway when it is opened on a cold windless day. The
larger density of the cold air produces a higher pressure on the outside of the
doorway which drives the cold air in at the bottom. If the temperature
difference is large enough you will experience a cool draft around your legs if
you stand in the doorway.
Gravity currents occur in gases when there are temperature
differences, as in the doorway flow just described. An important atmospheric
example is the sea breeze, which is the flow of cool moist air from the sea to
the land. On a warm day the sun heats the land more than the sea and,
consequently, the air at low altitudes over the land is warmer than that over
the sea. The resulting density difference drives the sea breeze. The sea breeze
is a significant feature of coastal meteorology in many parts of the world. For
example, the wind measured at Scripps Institution of Oceanography pier in
Another important class of gravity currents is the flow of
dense gases caused by the accidental release of a liquefied gas. There are many
examples of liquefied storage of gas. Chlorine, commonly used for sterilizing
swimming pools, is an example of a toxic gas that is stored in a pressurised
container. These containers are found in residential areas, and chlorine is
transported by road and rail. Flammable gases such as natural gas and propane
are also stored in this way, often in large quantities. If a leak occurs or the
container fails catastrophically, the released liquid vaporizes and produces
cold gas, which is denser than air because of its low temperature. Even for low
molecular weight gases such as methane the effects of temperature dominate and
the cold gas will, under most circumstances, produce a
gravity current. Because of the potential dangers of a toxic or
flammable gas spreading over the ground in populated regions, there has been
considerable research into the consequences of such accidental spills over the
past 20 years and much of our understanding of gravity currents comes from field,
laboratory and theoretical studies focused on this problem. A nice recent
summary of the subject is John Simpson’s book “Gravity currents in the
laboratory and the environment” published in 1997 by Cambridge University
Press.
We focus on the dynamics of gravity
currents and have been particularly concerned with currents produced by lock
exchange, where two fluids of different densities are initially separated by a
vertical barrier. The gravity currents that result when the barrier is removed
have been studied in a number of experiments. A summary of present theory,
essentially due to Benjamin (1968), can be found in [141] and we note that
there are some difficulties with his theory. Our research has examined the
assumptions behind Benjamin’s theory. In [147] we examine the flow in an
intrusion along an interface, which is an approximation of a current along a
stress-free boundary. We find that, in contrast to the accepted view, there is
little relative flow and that Benjamin’s assumption that the internal flow is
small is valid. In [153] we present experimental observations of lock exchange
currents that appear to be energy conserving in that they occupy half the
channel depth and have flow speeds close to that predicted by Benjamin. We also
study currents in deep ambient fluids and show that, in contrast to Benjamin’s
theory, they too can be described by a new energy-conserving theory. We show
that both von Karman’s (1940) and Benjamin’s (1968)
result that the front Froude number is √2 when the ambient is infinitely
deep is wrong and that the correct value is 1.
Recently, this work has been extended to
non-Boussinesq currents. When the density difference becomes large, there is an
asymmetry between the flow of the light and heavy current – see [141]. The
theory is singular in the limit of vanishing density ratio, and we have
recently carried out experiments which show that it is necessary for an
additional hydraulic jump to form. A theory, which includes the jump (but see
below) is under development. Some details of this and the Boussinesq case can
be found in the lecture I gave at the European Geophysical Society in April
2002 (see EGS_2002.pdf).
The effects of rotation on lock exchange currents are
discussed in [148]. This paper extends Benjamin’s full-depth lock exchange
theory to include the effects of rotation. energy-conserving
gravity currents in rotating channels.
The theory is an extension of Benjamin’s for nonrotating gravity
currents, and in a similar way makes use of the steady-state and prefect-fluid
(incompressible, inviscid, and immiscible) approximations, and supposes the
existence of a hydrostatic control point in the current some distance away from
the nose. The model allows for fully nonhydrostatic and ageostrophic motion in
a control volume V ahead of the control point, with the solution being
determined by the requirements that energy and momentum are conserved. The
governing parameter is the ratio of the channel width to the internal Rossby
radius of deformation. Analytic solutions are determined for the particular
case of zero front-relative flow within the gravity current.
2 Two-layer bores
A closely related flow to the gravity current is the
two-layer bore. Consider a dense gravity current travelling along the lower
boundary into a lighter ambient fluid. Now suppose that the ambient fluid ahead
of the current also has a lower layer of fluid with the same density as the
current. In this case the propagating disturbance is a bore and its speed
depends on the depth of the current and the depth of the layer ahead.
The classical case when the dense fluid is water and the
less dense fluid is air is observed in many rivers, such as the River Severn as
shown. The flow in this one-layer bore (or hydraulic jump if the bore is
stationary) is determined by the conservation of volume and momentum fluxes –
the analysis can be found in standard fluid dynamics text books. It is also
known that there is dissipation as the fluid passes through the jump.
For the two-layer case conservation of volume and momentum
fluxes are not sufficient to determine the flow. It is necessary to state where
the dissipation occurs and this is not known a priori. There have been a number
of attempts to resolve this issue, and we discuss them in [141]. In the Boussinesq
case of small density differences, we have extended the energy-conserving
gravity current theory and obtain a good agreement with experiments on bores
produced by gravity currents. A paper on this work is in preparation. However,
the extension of this theory to the non-Boussinesq case is unclear.
We are about to start (July 2002) a project funded by NSF in
collaboration with UCSB (PI’s PFL, JWR, Eckart Meiburg) to examine this problem. Experiments and theory
will be done at UCSD and compared with DNS calculations done at UCSB.