Energy Balances for Axisymmetric Gravity Currents in Homogeneous and Linearly Stratified Ambients

We analyse the exchange of energy for an axisymmetric gravity current, released instantaneously from a lock, propagating over a horizontal boundary at high Reynolds number. The study is relevant to flow in either a wedge or a full circular geometry. Attention is focused on effects due to a linear stratification in the ambient. The investigation uses both a one-layer shallow-water model and Navier–Stokes finite-difference simulations. There is fair agreement between these two approaches for the energy changes of the dense fluid (the current). The stratification enhances the accumulation of potential energy in the ambient and reduces the energy decay (dissipation) of the two-fluid system. The total energy of the axisymmetric current decays considerably faster with distance of propagation than for the two-dimensional counterpart

Gravity currents: a personal perspective.

Gravity currents, driven by horizontal differences in buoyancy, play a central role\ud in fluid mechanics, with numerous important natural and industrial applications.\ud The first quantitative, fluid-mechanical study of gravity currents, by von K´arm´an in\ud 1940, was carried out before the birth of this Journal; the next important theoretical\ud contribution was in 1968 by Brooke Benjamin, and appeared in this Journal more\ud than a decade after its birth. The present paper reviews some of the material that has\ud built on this auspicious start. Part of the fun and satisfaction of being involved in\ud this field is that its development has been based on both theoretical and experimental\ud contributions, which have at times been motivated and supported by field observations\ud and measurements

On gravity currents driven by constant fluxes of saline and particle-laden fluid in the presence of a uniform flow

Experiments are reported on the sustained release of saline and particle-laden fluid\ud into a long, but relatively narrow, flume, filled with fresh water. The dense fluid rapidly\ud spread across the flume and flowed away from the source: the motion was then essentially\ud two-dimensional. In the absence of a background flow in the flume, the motion\ud was symmetric, away from the source. However, in the presence of a background\ud flow the upstream speed of propagation was slowed and the downstream speed was\ud increased. Measurements of this motion are reported and, when the excess density was\ud due to the presence of suspended sediment, the distribution of the deposited particles\ud was also determined. Alongside this experimental programme, new theoretical models\ud of the motion were developed. These were based upon multi-layered depth-averaged\ud shallow-water equations, in which the interfacial drag and mixing processes were\ud explicitly modelled. While the early stages of the motion are independent of these\ud interfacial phenomena to leading order, they play an increasingly important dynamical\ud role as the the flow is slowed, or even arrested. In addition a new integral model is\ud proposed. This does not resolve the interior dynamics of the flow, but may be readily\ud integrated and obviates the need for more lengthy numerical calculations. It is shown\ud that the predictions from both the shallow-layer and integral models are in close\ud agreement with the experimental observations

Leakage from gravity currents in a porous medium. Part 2. A line sink

We consider the propagation of a buoyancy-driven gravity current in a porous medium bounded by a horizontal, impermeable boundary. The current is fed by a constant flux injected at a point and leaks through a line sink at a distance from the injection point. This is an idealized model of how a fault in a cap rock might compromise the geological sequestration of carbon dioxide. The temporal evolution of the efficiency of storage, defined as the instantaneous ratio of the rate at which fluid is stored without leaking to the rate at which it is injected, is of particular interest. We show that the ‘efficiency of storage’ decays like t−2/5 for times t that are long compared with the time taken for the current to reach the fault. This algebraic decay is in contrast to the case of leakage through a circular sink (Neufeld et al., J. Fluid Mech., vol. 2010) where the efficiency of storage decays more slowly like 1/lnt. The implications of the predicted decay in the efficiency of storage are discussed in the context of geological sequestration of carbon dioxide. Using parameter values typical of the demonstration project at Sleipner, Norway, we show that the efficiency of storage should remain greater than 90% on a time scale of millennia, provided that there are no significant faults in the cap rock within about 12km of the injection site

Two-phase gravity currents in porous media

We develop a model describing the buoyancy-driven propagation of two-phase gravity currents, motivated by problems in groundwater hydrology and geological storage of carbon dioxide (CO2). In these settings, fluid invades a porous medium saturated with an immiscible second fluid of different density and viscosity. The action of capillary forces in the porous medium results in spatial variations of the saturation of the two fluids. Here, we consider the propagation of fluid in a semi-infinite porous medium across a horizontal, impermeable boundary. In such systems, once the aspect ratio is large, fluid flow is mainly horizontal and the local saturation is determined by the vertical balance between capillary and gravitational forces. Gradients in the hydrostatic pressure along the current drive fluid flow in proportion to the saturation-dependent relative permeabilities, thus determining the shape and dynamics of two-phase currents. The resulting two-phase gravity current model is attractive because the formalism captures the essential macroscopic physics of multiphase flow in porous media. Residual trapping of CO2 by capillary forces is one of the key mechanisms that can permanently immobilize CO2 in the societally important example of geological CO2 sequestration. The magnitude of residual trapping is set by the areal extent and saturation distribution within the current, both of which are predicted by the two-phase gravity current model. Hence the magnitude of residual trapping during the post-injection buoyant rise of CO2 can be estimated quantitatively. We show that residual trapping increases in the presence of a capillary fringe, despite the decrease in average saturation

Instability of a gravity current within a soap film

One of the simplest geometries in which to study fluid flow between two soap films connected by a Plateau border is provided by a catenoid with a secondary film at its narrowest point. Dynamic variations in the spacing between the two rings supporting the catenoid lead to fluid flow between the primary and secondary films. When the rings are moved apart, while keeping their spacing within the overall stability regime of the films, after a rapid thickening of the secondary film the excess fluid in it starts to drain into the sloped primary film through the Plateau border at which they meet. This influx of fluid is accommodated by a local thickening of the primary film. Experiments described here show that after this drainage begins the leading edge of the gravity current becomes linearly unstable to a finite-wavelength fingering instability. A theoretical model based on lubrication theory is used to explain the mechanism of this instability. The predicted characteristic wavelength of the instability is shown to be in good agreement with experimental results. Since the gravity current advances into a film of finite, albeit microscopic, thickness this situation is one in which the regularization often invoked to address singularities at the nose of a thin film is physically justified

Stokes at 200: A celebration of the remarkable achievements of Sir George Gabriel Stokes two hundred years after his birth

Sir George Gabriel Stokes PRS was for 30 years an inimitable Secretary of the Royal Society and its President from 1885 to 1890. Two hundred years after his birth, Stokes is a towering figure in physics and applied mathematics; fluids, asymptotics, optics, acoustics among many other fields. At the Stokes200 meeting, held at Pembroke College, Cambridge from 15-18th September 2019, an invited audience of about 100 discussed the state of the art in all the modern research fields that have sprung from his work in physics and mathematics, along with the history of how we have got from Stokes' contributions to where we are now. This theme issue is based on work presented at the Stokes200 meeting. In bringing together people whose work today is based upon Stokes' own, we aim to emphasize his influence and legacy at 200 to the community as a whole. This article is part of the theme issue 'Stokes at 200 (Part 1)'

Stokes at 200 (part 2)

We present the second half of the papers from the Stokes200 symposium celebrating the bicentenary of George Gabriel Stokes. This article is part of the theme issue 'Stokes at 200 (part 2)'

Amplitude equations for a system with thermohaline convection

The multiple scale expansion method is used to derive amplitude equations for a system with thermohaline convection in the neighborhood of Hopf and Taylor bifurcation points and at the double zero point of the dispersion relation. A complex Ginzburg-Landau equation, a Newell-Whitehead-type equation, and an equation of the $\phi^4$ type, respectively, were obtained. Analytic expressions for the coefficients of these equations and their various asymptotic forms are presented. In the case of Hopf bifurcation for low and high frequencies, the amplitude equation reduces to a perturbed nonlinear Schr\"odinger equation. In the high-frequency limit, structures of the type of "dark" solitons are characteristic of the examined physical system.Comment: 21 pages, 8 figure

Lava channel formation during the 2001 eruption on Mount Etna: evidence for mechanical erosion

We report the direct observation of a peculiar lava channel that was formed near the base of a parasitic cone during the 2001 eruption on Mount Etna. Erosive processes by flowing lava are commonly attributed to thermal erosion. However, field evidence strongly suggests that models of thermal erosion cannot explain the formation of this channel. Here, we put forward the idea that the essential erosion mechanism was abrasive wear. By applying a simple model from tribology we demonstrate that the available data agree favorably with our hypothesis. Consequently, we propose that erosional processes resembling the wear phenomena in glacial erosion are possible in a volcanic environment.Comment: accepted for publication in Physical Review Letter