Gravity current

In fluid dynamics, a gravity current is a primarily horizontal flow in a gravitational field that is driven by a density difference, hence gravity currents also sometimes being referred to as "density currents". Typically, the density difference is small enough for the Boussinesq approximation to be valid.

Gravity currents are typically of very low aspect ratio (that is, height over typical horizontal lengthscale). The pressure distribution is thus approximately hydrostatic, apart from near the leading edge (this may be seen using dimensional analysis). Thus gravity currents may be simulated by the shallow water equations, with special dispensation for the leading edge which behaves as a discontinuity.

Structure

A typical gravity current consists of a head and tail structure. The head, which is the leading edge of the gravity current, is a region in which relatively large volumes of ambient fluid are displaced. The tail is the bulk of fluid which follows the head and which is best though of as the fluid which is left behind as the gravity current advances.

Immediately in the wake of the head intense mixing occurs between the gravity current and the ambient fluid occurs. Mixing occurs from both above and below the gravity current. Mixing from above is a result of turbulent billows (Kelvin-Helmholtz instabilities) which in the wake of the head and engulf ambient fluid into the tail, a process referred to as "entrainment". Mixing from below is a result of the gravity current overrunning ambient fluid, literally trapping it underneath it. Direct mixing also occurs at the front of the head through lobes and cleft structures which form on the surface of the head. According to one paradigm, the leading edge of a gravity current 'controls' the flow behind it: it provides a boundary condition for the flow.

The absorbed ambient fluid mixes with the tail of the gravity current, which means that the head of the current leaves behind it a layer of less dense fluid. In effect this can be though of like a rocket which leaves a streak of fumes behind it as it advances. In terms of structure, the tail consists of two counterflowing currents of fluid. At the bottom dense fluid flows towards the head. It is this dense current which drives the current, providing new driving head to replace the head lost due to entrainment. Above this dense current is a counterflowing current of less dense fluid which is a result of dense fluid mixing with lighter ambient fluid at the head.

Propagation

The leading edge moves at a Froude number of about 1; estimates of the exact value vary between about 0.7 and 1.4.

Gravity currents can originate either from finite releases or from constant releases. In the case of constant releases, the fluid in the head is constantly replaced and the gravity current can therefore propagate, in theory, for ever. In practice, constant releases could be encountered at river estuaries, where fresh water of low density encounters denser sea water and at high tide, the sea is pushing into the estuary. The sea water thereby constitutes a theoretically constant release gravity current. Of course, the tide will at some stage reverse and the gravity current thereby dissipate.

Most gravity currents will in fact occur as a result of a finite-volume release of fluid. In this case the propagation usually occurs in three phases. In the first phase the gravity current propagation is turbulent. The flow displays the billowing patterns described above and much mixing between the current and the environment can be expected. In this phase the propagation rate of the current is approximately constant with time.^[1]

As the driving fluid depletes as a result of the current spreading into the environment, the driving head decreases until the flow becomes laminar. In this phase, there is only very little mixing and the billowing structure of the flow disappears. From this phase onwards the propagation rate decreases with time and the current gradually slows down.

Finally, as the current spreads even further, it becomes so thin that viscous forces between the intruding fluid and the ambient and boundaries govern the flow. In this phases no more mixing occurs and the propagation rate slows down even more.^[1][2]

The spread of a gravity current depends on the boundary conditions, and two cases are usually distinguished depending on whether the initial release is of the same with as the environment or not.

In the case where the widths are the same, one obtains what is usually referred to as a "lock-exchange" or a "corridor" flow. This refers to the flow spreading along walls on both sides and effectively keeping a constant width whilst it propagates. In this case the flow is effectively two-dimensional. Experiments on variations of this flow have been made with lock-exchange flows propagating in narrowing/expanding environments. Effectively, a narrowing environment will result in the depth of the head increasing as the current advances and thereby its rate of propagation increasing with time, whilst in an expanding environment the opposite will occur.^{[citation needed]}

In the other case, the flow spreads radially from the source forming an "axisymmetric" flow. The angle of spread depends on the release conditions. In the case of a point release, an extremely rare event in nature, the spread is perfectly axisymmetric, in all other cases the current will for a sector.

When a gravity encounters a solid boundary, it can either overcome the boundary, by flowing around or over it, or be reflected by it. The actual outcome of the collision depends primarily on the height and width of the obstacle. If the obstacle is shallow (part) of the gravity current will overcome the obstacle by flowing over it. Similarly, if the width of the obstacle is small, the gravity current will flow around it, just like a river flows around a boulder.

If the obstacle cannot be overcome, provided propagation is in the turbulent phase, the gravity current will first surge vertically up (or down depending on the density contrast) along the obstacle, a process known as "sloshing". Sloshing induces a lot of mixing between the ambient and the current and this forms an accumulation of lighter fluid against the obstacle. As more and more fluid accumulates against the obstacle, this starts to propagate in the opposite direction to the initial current, effectively resulting in a second gravity current flowing on top of the original gravity current. This reflection process is a common feature of doorway flows (see below), where a gravity current flows into a finite-size space. In this case the flow repeatedly collides with the end walls of the space, causing a series of currents travelling back and forth between opposite walls. This process has been described in details by Lane-Serf.^[3]

Research

Due to their ubiquitousness in nature gravity currents have been and still are intensely studied in laboratories all over the world.

The first mathematical study of the propagation of gravity currents can be attributed to T. B. Benjamin.^[4] Observations of intrusions and collisions between fluids of differing density were made well before T. B. Benjamin's study, see for example by M. B. Abbot^[5] or D. I. H. Barr.^[6]

J. E. Simpson from the Department of Applied Mathematics and Theoretical Physics of Cambridge University in the UK has carried out longstanding research on gravity currents and issued a multitude of papers on the subject. He published an article^[7] in 1982 for Annual Review of Fluid Mechanics which summarises the state of research in the domain of gravity currents at the time. Although now nearly 30 years old, his article forms a good introduction to the subject. Simpson also published a more detailed book on the topic.^[8]

In nature and the built environment

Gravity currents are capable of transporting material across large horizontal distances. For example, turbidity currents on the seafloor may carry material thousands of kilometres.

Gravity currents occur at a variety of scales throughout nature. Examples include oceanic fronts, avalanches, haboobs, seafloor turbidity currents, lahars, pyroclastic flows, and lava flows. There are also gravity currents with large density variations - the so-called low Mach number compressible flows. An example of such a gravity current is the heavy gas dispersion in the atmosphere with initial ratio of gas density to density of atmosphere about 1.5-5.

Gravity currents are frequently encountered in the built environment in the form of doorway flows. These occur when a door (or window) separates two rooms of different temperature and air exchanges are allowed to occur. This can for example be experienced when sitting in a heated lobby during winter and the entrance door is suddenly opened. In this case the cold air will first be felt by ones feet as a result of the outside air propagating as a gravity current along the floor of the room. Doorway flows are of interest in the domain of natural ventilation and air conditioning/refrigeration and have been extensively investigated, see for example Kiel and Wilson,^[9] Dalziel and Lane-Serff^[10] or Phillips and Woods,^[11] to name but a few.

References

1. ^ ^{a b} Huppert, H. E.; Simpson, J. E. (1980). "The Slumping of Gravity Currents". Journal of Fluid Mechanics 99 (04): 785–799. Bibcode 1980JFM....99..785H. doi:10.1017/S0022112080000894. 
2. ^ Fay, J. A. (1969). "The Spread of Oil Slicks on a Calm Sea". In Hoult, D. P. Oil on the Sea. 
3. ^ Lane-Serf, G. F. (1989). "Heat Flow and Air Movement in Buildings". PhD Thesis (University of Cambridge). 
4. ^ Benjamin, T. B. (1968). "Gravity Current and Related Phenomena". Journal of Fluid Mechanics 31: 209–248. Bibcode 1968JFM....31..209B. doi:10.1017/S0022112068000133. 
5. ^ Abbot, M. B. (1961). "On the Spreading of one Fluid Over Another. Part II: The Wave Front". La Houille Blanche 6: 827–836. 
6. ^ Barr, D. I. H. (1967). "Densimetric Exchange Flows in Rectangular Channels". La Houille Blanche 22: 619–631. 
7. ^ Simpson, J. E. (1982). "Gravity Currents in the Laboratory, Atmosphere, and Ocean". Annual Review of Fluid Mechanics 14: 213–234. Bibcode 1982AnRFM..14..213S. doi:10.1146/annurev.fl.14.010182.001241. 
8. ^ Simpson, J. E (1999). Gravity Currents: In the Environment and the Laboratory. Cambridge University Press. 
9. ^ Kiel, D. E.; Wilson, D. J. (1990). "Gravity Driven Counter Flow Through an Open Door in a Sealed Room". Building and Environment 25 (4): 379–388. doi:10.1016/0360-1323(90)90012-G. 
10. ^ Dalziel, S. B.; Lane-Serff, G. F. (1991). "The Hydraulics of Doorway Exchange Flows". Building and Environment 26 (2): 121–135. doi:10.1016/0360-1323(91)90019-8. 
11. ^ Phillips, J. C.; Woods, A. W. (2004). "On Ventilation of a Heated Room through a Single Doorway". Building and Environment 39: 241–253. doi:10.1016/j.buildenv.2003.09.002.