Propagation of viscous currents on a porous substrate with finite capillary entry pressure

2016 ◽  
Vol 801 ◽  
pp. 65-90 ◽  
Author(s):  
Roiy Sayag ◽  
Jerome A. Neufeld
Keyword(s):  
Laboratory Experiments ◽  
Gravity Current ◽  
Fluid Pressure ◽  
Analytical Techniques ◽  
Gravity Currents ◽  
Porous Substrate ◽  
Entry Pressure ◽  
Self Similar ◽  
Capillary Entry Pressure ◽  
Theoretical Results

We study the propagation of viscous gravity currents over a thin porous substrate with finite capillary entry pressure. Near the origin, where the current is deep, propagation of the current coincides with leakage through the substrate. Near the nose of the current, where the current is thin and the fluid pressure is below the capillary entry pressure, drainage is absent. Consequently the flow can be characterised by the evolution of drainage and fluid fronts. We analyse this flow using numerical and analytical techniques combined with laboratory-scale experiments. At early times, we find that the position of both fronts evolve as $t^{1/2}$, similar to an axisymmetric gravity current on an impermeable substrate. At later times, the growing effect of drainage inhibits spreading, causing the drainage front to logarithmically approach a steady position. In contrast, the asymptotic propagation of the fluid front is quasi-self-similar, having identical structure to the solution of gravity currents on an impermeable substrate, only with slowly varying fluid flux. We benchmark these theoretical results with laboratory experiments that are consistent with our modelling assumption, but that also highlight the detailed dynamics of drainage inhibited by finite capillary pressure.

Converging gravity currents over a permeable substrate

2015 ◽  
Vol 778 ◽  
pp. 669-690 ◽  
Author(s):  
Zhong Zheng ◽  
Sangwoo Shin ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Gravity Current ◽  
Gravity Currents ◽  
Numerical Predictions ◽  
Front Position ◽  
Dependent Position ◽  
Self Similar ◽  
Vertical Leakage ◽  
Similar Solutions ◽  
Lock Gate

We study the propagation of viscous gravity currents along a thin permeable substrate where slow vertical drainage is allowed from the boundary. In particular, we report the effect of this vertical fluid drainage on the second-kind self-similar solutions for the shape of the fluid–fluid interface in three contexts: (i) viscous axisymmetric gravity currents converging towards the centre of a cylindrical container; (ii) viscous gravity currents moving towards the origin in a horizontal Hele-Shaw channel with a power-law varying gap thickness in the horizontal direction; and (iii) viscous gravity currents propagating towards the origin of a porous medium with horizontal permeability and porosity gradients in power-law forms. For each of these cases with vertical leakage, we identify a regime diagram that characterizes whether the front reaches the origin or not; in particular, when the front does not reach the origin, we calculate the final location of the front. We have also conducted laboratory experiments with a cylindrical lock gate to generate a converging viscous gravity current where vertical fluid drainage is allowed from various perforated horizontal substrates. The time-dependent position of the propagating front is captured from the experiments, and the front position is found to agree well with the theoretical and numerical predictions when surface tension effects can be neglected.

Self-similar viscous gravity currents: phase-plane formalism

1990 ◽  
Vol 210 ◽  
pp. 155-182 ◽  
Author(s):  
Julio Gratton ◽  
Fernando Minotti
Keyword(s):  
Scaling Laws ◽  
Phase Plane ◽  
Gravity Current ◽  
Classical Problem ◽  
Gravity Currents ◽  
Steady Flows ◽  
Two Phase ◽  
Integral Curves ◽  
Self Similar ◽  
Similar Solutions

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

Gravity currents over fractured substrates in a porous medium

2007 ◽  
Vol 584 ◽  
pp. 415-431 ◽  
Author(s):  
DAVID PRITCHARD
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Gravity Current ◽  
Gravity Currents ◽  
Limited Extent ◽  
Form Part ◽  
Long Time ◽  
Self Similar ◽  
Steady State Solutions ◽  
A Current

We consider the behaviour of a gravity current in a porous medium when the horizontal surface along which it spreads is punctuated either by narrow fractures or by permeable regions of limited extent. We derive steady-state solutions for the current, and show that these form part of a long-time asymptotic description which may also include a self-similar ‘leakage current’ propagating beyond the fractured region with a length proportional to t1/2. We discuss the conditions under which a current can be completely trapped by a permeable region or a series of fractures.

scholarly journals Lubricated viscous gravity currents

2015 ◽  
Vol 766 ◽  
pp. 626-655 ◽  
Author(s):  
Katarzyna N. Kowal ◽  
M. Grae Worster
Keyword(s):  
Gravity Current ◽  
Gravity Currents ◽  
Singular Limit ◽  
Vertical Shear ◽  
Surface Films ◽  
Viscous Coupling ◽  
Self Similarity ◽  
Transient Behaviour ◽  
Fingering Instability ◽  
Self Similar

AbstractWe present a theoretical and experimental study of viscous gravity currents lubricated by another viscous fluid from below. We use lubrication theory to model both layers as Newtonian fluids spreading under their own weight in two-dimensional and axisymmetric settings over a smooth rigid horizontal surface and consider the limit in which vertical shear provides the dominant resistance to the flow in both layers. There are contributions from Poiseuille-like flow driven by buoyancy and Couette-like flow driven by viscous coupling between the layers. The flow is self-similar if both fluids are released simultaneously, and exhibits initial transient behaviour when there is a delay between the initiation of flow in the two layers. We solve for both situations and show that the latter converges towards self-similarity at late times. The flow depends on three key dimensionless parameters relating the relative dynamic viscosities, input fluxes and density differences between the two layers. Provided the density difference between the two layers is bounded away from zero, we find an asymptotic solution in which the front of the lubricant is driven by its own gravitational spreading. There is a singular limit of equal densities in which the lubricant no longer spreads under its own weight in the vicinity of its nose and ends abruptly with a non-zero thickness there. We explore various regimes, from thin lubricating layers underneath a more viscous current to thin surface films coating an underlying more viscous current and find that although a thin film does not greatly influence the more viscous current if it forms a surface coating, it begins to cause interesting dynamics if it lubricates the more viscous current from below. We find experimentally that a lubricated gravity current is prone to a fingering instability.

The effects of drag on turbulent gravity currents

2000 ◽  
Vol 416 ◽  
pp. 297-314 ◽  
Author(s):  
LYNNE HATCHER ◽  
ANDREW J. HOGG ◽  
ANDREW W. WOODS
Keyword(s):  
Analytical Solution ◽  
Laboratory Experiments ◽  
Gravity Current ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Flow Speed ◽  
Series Solutions ◽  
Accurate Approximation ◽  
New Class ◽  
Power Series Solutions

We model the propagation of turbulent gravity currents through an array of obstacles which exert a drag force on the flow proportional to the square of the flow speed. A new class of similarity solutions is constructed to describe the flows that develop from a source of strength q0tγ. An analytical solution exists for a finite release, γ = 0, while power series solutions are developed for sources with γ > 0. These are shown to provide an accurate approximation to the numerically calculated similarity solutions. The model is successfully tested against a series of new laboratory experiments which investigate the motion of a turbulent gravity current through a large flume containing an array of obstacles. The model is extended to account for the effects of a sloping boundary. Finally, a series of geophysical and environmental applications of the model are discussed.

scholarly journals Gravity current propagation up a valley

2014 ◽  
Vol 762 ◽  
pp. 417-434 ◽  
Author(s):  
Catherine S. Jones ◽  
Claudia Cenedese ◽  
Eric P. Chassignet ◽  
P. F. Linden ◽  
Bruce R. Sutherland
Keyword(s):  
Numerical Simulations ◽  
Laboratory Experiments ◽  
Rectangular Channel ◽  
Gravity Current ◽  
Gravity Currents ◽  
Momentum Transport ◽  
Front Speed

AbstractThe advance of the front of a dense gravity current propagating in a rectangular channel and V-shaped valley both horizontally and up a shallow slope is examined through theory, full-depth lock–release laboratory experiments and hydrostatic numerical simulations. Consistent with theory, experiments and simulations show that the front speed is relatively faster in the valley than in the channel. The front speed measured shortly after release from the lock is 5–22 % smaller than theory, with greater discrepancy found in upsloping V-shaped valleys. By contrast, the simulated speed is approximately 6 % larger than theory, showing no dependence on slope for rise angles up to ${\it\theta}=8^{\circ }$. Unlike gravity currents in a channel, the current head is observed in experiments to be more turbulent when propagating in a V-shaped valley. The turbulence is presumably enhanced due to the lateral flows down the sloping sides of the valley. As a consequence, lateral momentum transport contributes to the observed lower initial speeds. A Wentzel–Kramers–Brillouin like theory predicting the deceleration of the current as it runs upslope agrees remarkably well with simulations and with most experiments, within errors.

Gravity currents over porous substrates

1998 ◽  
Vol 366 ◽  
pp. 239-258 ◽  
Author(s):  
L. P. THOMAS ◽  
B. M. MARINO ◽  
P. F. LINDEN
Keyword(s):  
Time Constant ◽  
Initial Conditions ◽  
Rectangular Cross Section ◽  
Scaling Law ◽  
Gravity Current ◽  
Gravity Currents ◽  
Porous Substrate ◽  
Homogeneous Fluid ◽  
Front Position ◽  
Loss Of Mass

Results of laboratory experiments are presented in which a fixed volume of homogeneous fluid is suddenly released into another fluid of slightly lower density, over a horizontal thin metallic grid placed a given distance above the solid bottom of a rectangular-cross-section channel. Dense liquid develops as a gravity current over the grid at the same time as it partially flows downwards. The results show that the gravity current loses mass at an exponential rate through the porous substrate with a time constant τ; the front velocity and the head of the current also decrease exponentially. The loss of mass dominates the flow and, in contrast to gravity currents running over solid bottoms, no self-similar inertial regime seems to be developed. A simple model is introduced to explain the scaling law of the loss of mass and the evolution of the front position. The flow evolution depends on the characteristic time of the initial (slumping) phase and the time constant τ, related to the initial conditions and the permeability of the porous substrate, respectively. Qualitative comparisons with other gravity currents with loss of mass, such as particle-driven gravity currents, are provided.

scholarly journals Dynamics of viscous grounding lines

2010 ◽  
Vol 648 ◽  
pp. 363-380 ◽  
Author(s):  
ROSALYN A. V. ROBISON ◽  
HERBERT E. HUPPERT ◽  
M. GRAE WORSTER
Keyword(s):  
Laboratory Experiments ◽  
Gravity Current ◽  
Ice Sheets ◽  
Gravity Currents ◽  
Solid Base ◽  
Leading Order ◽  
Ice Shelves ◽  
Grounding Line ◽  
Theoretical Predictions ◽  
Good Agreement

We have used viscous fluids in simple laboratory experiments to explore the dynamics of grounding lines between marine ice sheets and the freely floating ice shelves into which they develop. We model the ice sheets as shear-dominated gravity currents, and the ice shelves as extensional gravity currents having zero shear to leading order. We consider the flow of viscous fluid down an inclined plane into a dense inviscid ‘ocean’. A fixed flux of fluid is supplied at the top of the plane, which is at ‘sea level’. The fluid forms a gravity current flowing down and attached to the plane for some distance before detaching to form a freely floating extensional current. We have derived a mathematical model of the flow that incorporates a new dynamic boundary condition for the position of the grounding line, where the gravity current loses contact with the solid base. The grounding line initially advances and eventually reaches a steady position. Good agreement between our theoretical predictions and experimental measurements and observations gives confidence in the fundamental assumptions of our model.

A study of three-dimensional gravity currents on a uniform slope

2002 ◽  
Vol 453 ◽  
pp. 239-261 ◽  
Author(s):  
ANDREW N. ROSS ◽  
P. F. LINDEN ◽  
STUART B. DALZIEL
Keyword(s):  
Laboratory Experiments ◽  
Gravity Current ◽  
Three Dimensional ◽  
Gravity Currents ◽  
Integral Model ◽  
Conductivity Measurements ◽  
Dense Fluid ◽  
Wedge Shape ◽  
Dimensional Gravity ◽  
Shallow Water Models

In many geophysical, environmental and industrial situations, a finite volume of fluid with a density different to the ambient is released on a sloping boundary. This leads to the formation of a gravity current travelling up, down and across the slope. We present novel laboratory experiments in which the dense fluid spreads both down-slope (and initially up-slope) and laterally across the slope. The position, shape and dilution of the current are determined through video and conductivity measurements for moderate slopes (5° to 20°). The entrainment coefficient for different slopes is calculated from the experimental results and is found to depend very little on the slope. The value agrees well with previously published values for entrainment into gravity currents on a horizontal surface. The experimental measurements are compared with previous shallow-water models and with a new wedge integral model developed and presented here. It is concluded that these simplified models do not capture all the significant features of the flow. In the models, the current takes the form of a wedge which travels down the slope, but the experiments show the formation of a more complicated current. It is found that the wedge integral model over-predicts the length and width of the gravity current but gives fair agreement with the measured densities in the head. The initial stages of the flow, during which time the wedge shape develops, are studied. It is found that although the influence of the slope is seen relatively quickly for moderate slopes, the time taken for the wedge to develop is much longer. The implications of these findings for safety analysis are briefly discussed.

Non-self-similar viscous gravity currents

2018 ◽  
Vol 3 (3) ◽  
Author(s):  
Bruce R. Sutherland ◽  
Kristen Cote ◽  
Youn Sub (Dominic) Hong ◽  
Luke Steverango ◽  
Chris Surma
Keyword(s):  
Gravity Currents ◽  
Self Similar
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