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

Shallow current of viscous fluid flowing between diverging or converging walls

We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power-law ($x^n$), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$, where $t$ is time and $\alpha$ is a constant. The invading fluid has higher viscosity than the ambient fluid, thus avoiding Saffman-Taylor instability. Similarity solutions of the first kind for the outflow problem are found using approximations of lubrication theory. Zheng et al [2014] studied the deep-channel case and found divergent behaviour of the similarity variable as $n\rightarrow1$ and $n\rightarrow3$, when fluid flows into and out of the channel respectively. No divergence is found in the shallow case presented here. The characteristic equilibration time for the numerically simulated constant-volume flow to converge to the similarity solution is calculated assuming inverse dependence on the ratio disagreement between the current front using the method of lines (MOL). The inverse power dependence between equilibration time and ratio disagreement is found for channels of different powers. A similarity solution of the second kind for the inflow problem is found using the phase plane formalism and the bisection method. An exponential decay relationship is found between $n$ and the degree $\delta$ of the similarity variable $xt^{-\delta}$, which does not show any divergent behaviour for large $n$. An asymptotic behaviour is found for $\delta$ that approaches $1/2$ as $n\rightarrow\infty$

A 16-channel Digital TDC Chip with internal buffering and selective readout for the DIRC Cherenkov counter of the BABAR experiment

A 16-channel digital TDC chip has been built for the DIRC Cherenkov counter of the BaBar experiment at the SLAC B-factory (Stanford, USA). The binning is 0.5 ns, the conversion time 32 ns and the full-scale 32 mus. The data driven architecture integrates channel buffering and selective readout of data falling within a programmable time window. The time measuring scale is constantly locked to the phase of the (external) clock. The linearity is better than 80 ps rms. The dead time loss is less than 0.1% for incoherent random input at a rate of 100 khz on each channel. At such a rate the power dissipation is less than 100 mw. The die size is 36 mm2.Comment: Latex, 18 pages, 13 figures (14 .eps files), submitted to NIM

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

On Nichols algebras over PGL(2,q) and PSL(2,q)

We compute necessary conditions on Yetter-Drinfeld modules over the groups \mathbf{PGL}(2,q)=\mathbf{PGL}(2,\FF_q) and \mathbf{PSL}(2,q)=\mathbf{PSL}(2,\FF_q) to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that there is no non-trivial finite-dimensional pointed Hopf algebra over the Mathieu groups $M_{20}$ and $M_{21}=\mathbf{PSL}(3,4)$.Comment: Minor change

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

Simply connected projective manifolds in characteristic p>0p>0 have no nontrivial stratified bundles

We show that simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles. This gives a positive answer to a conjecture by D. Gieseker. The proof uses Hrushovski's theorem on periodic points.Comment: 16 pages. Revised version contains a more general theorem on torsion points on moduli, together with an illustration in rank 2 due to M. Raynaud. Reference added. Last version has some typos corrected. Appears in Invent.math