Solutions near the hydraulic control point in a gravity current model

1995 ◽  
Vol 7 (12) ◽  
pp. 2972-2977 ◽  
Author(s):  
G. Alendal
Keyword(s):  
Current Model ◽  
Control Point ◽  
Gravity Current ◽  
Hydraulic Control

A bottom gravity current model for CO2-enriched seawater

1993 ◽  
Vol 34 (9-11) ◽  
pp. 1065-1072 ◽  
Author(s):  
Helge Drange ◽  
Guttorm Alendal ◽  
Peter M. Haugan
Keyword(s):  
Current Model ◽  
Gravity Current

scholarly journals Viscous effects in two-layer, unidirectional hydraulic flow

2010 ◽  
Vol 644 ◽  
pp. 371-394
Author(s):  
MARTIN S. SINGH ◽  
ANDREW McC. HOGG
Keyword(s):  
Control Point ◽  
Lower Boundary ◽  
Internal Gravity Waves ◽  
Velocity Shear ◽  
Hydraulic Control ◽  
Viscous Effects ◽  
Unidirectional Flow ◽  
Boundary Case ◽  
Flow Through ◽  
Layer Flow

Hydraulic equations are derived for a stratified (two-layer) flow in which the horizontal velocity varies continuously in the vertical. Viscosity is included in the governing equations, and the effect of friction in hydraulically controlled flows is examined. The analysis yields Froude numbers which depend upon the integrated inverse square of velocity but reduce to the original layered Froude numbers when velocity is constant with depth. The Froude numbers reveal a critical condition for hydraulic control, which equates to the arrest of internal gravity waves.Solutions are presented for the case of unidirectional flow through a lateral constriction, both with and without bottom drag. In the free-slip lower boundary case, viscosity transports momentum from the faster to the slower layer, thereby shifting the control point downstream and reducing the flux through the constriction. However, while the velocity shear at the interface between the two layers is reduced, the top-to-bottom velocity difference of the controlled solution is increased for larger values of viscosity. This counter-intuitive result is due to the restrictions placed on the flow at the hydraulic control point. When bottom drag is included in the model, the total flux may increase, in some cases exceeding that of the inviscid solution.

Two-phase gravity currents in porous media

2011 ◽  
Vol 678 ◽  
pp. 248-270 ◽  
Author(s):  
MADELEINE J. GOLDING ◽  
JEROME A. NEUFELD ◽  
MARC A. HESSE ◽  
HERBERT E. HUPPERT
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Current Model ◽  
Gravity Current ◽  
Capillary Forces ◽  
Gravity Currents ◽  
Flow In Porous Media ◽  
Two Phase ◽  
Residual Trapping

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.

Buoyant flow of through and around a semi-permeable layer of finite extent

2016 ◽  
Vol 809 ◽  
pp. 553-584 ◽  
Author(s):  
Tri Dat Ngo ◽  
Emmanuel Mouche ◽  
Pascal Audigane
Keyword(s):  
Capillary Pressure ◽  
Graphical Representation ◽  
Current Model ◽  
Gravity Current ◽  
Two Dimensions ◽  
Total Flux ◽  
Vertical Column ◽  
Fine Grid ◽  
Permeable Layer ◽  
Counter Current

The buoyancy- and capillary-driven counter-current flow of $\text{CO}_{2}$ and brine through and around a semi-permeable layer is studied both numerically and theoretically. The continuities of the capillary pressure and the total flux at the interface between the permeable matrix and layer control the $\text{CO}_{2}$ saturation discontinuity at the interface and the balance between the buoyant and capillary diffusion fluxes on each side of the interface. This interface process is first studied in a one-dimensional (1-D) vertical column geometry using the concept of extended capillary pressure and a graphical representation of the continuity conditions in the ($S_{L}$, $S_{U}$) plane, where $S_{L}$ and $S_{U}$ are the lower and upper saturation traces at the interface, respectively. In two dimensions, we heuristically extend the two-phase gravity current model to the case where the current is bounded by a semi-permeable layer. Consequently, the current is not saturated with $\text{CO}_{2}$, and its saturation and shape are derived from the flux and capillary pressure continuity conditions at the interface. This simplified model, which depends on $\text{CO}_{2}$ saturation only, is compared to fine grid simulations in the capillary-free and gravity-dominant cases. A good agreement is obtained in the second case; the current geometrical characteristics are accurately described. In the capillary-free case, we demonstrate that the local total velocity, which is, on average, zero because the flow is counter-current, must be considered in the total flux at the interface to obtain the same level of agreement.

Gravity current flow over obstacles

1995 ◽  
Vol 292 ◽  
pp. 39-53 ◽  
Author(s):  
G. F. Lane-Serff ◽  
L. M. Beal ◽  
T. D. Hadfield
Keyword(s):  
Theoretical Analysis ◽  
Hydraulic Jump ◽  
Lower Layer ◽  
Surface Condition ◽  
Gravity Current ◽  
Finite Depth ◽  
Return Flow ◽  
Hydraulic Control ◽  
Sea Breezes ◽  
Inflow Conditions

When a gravity current meets an obstacle a proportion of the flow may continue over the obstacle while the rest is reflected back as a hydraulic jump. There are many examples of this type of flow, both in the natural and man-made environment (e.g. sea breezes meeting hills, dense gas and liquid releases meeting containment walls). Two-dimensional currents and obstacles, where the reflected jump is in the opposite direction to the incoming current, are examined by laboratory experiment and theoretical analysis. The investigation concentrates on the case of no net flow, so that there is a return flow in the (finite depth) upper layer. The theoretical analysis is based on shallow-water theory. Both a rigid lid and a free surface condition for the top of the upper layer are considered. The flow may be divided into several regions: the inflow conditions, the region around the hydraulic jump, the flow at the obstacle and the flow downstream of the obstacle. Both theoretical and empirical inflow conditions are examined; the jump conditions are based on assuming that the energy dissipation is confined to the lower layer; and the flow over the obstacle is described by hydraulic control theory. The predictions for the proportion of the flow that continues over the obstacle, the speed of the reflected jump and the depth of the reflected flow are compared with the laboratory experiments, and give reasonable agreement. A shallower upper layer (which must result in a faster return velocity in the upper layer) is found to have a significant effect, both on the initial incoming gravity current and on the proportion of the flow that continues over the obstacle.

Gravity currents in rotating channels. Part 1. Steady-state theory

2002 ◽  
Vol 457 ◽  
pp. 295-324 ◽  
Author(s):  
J. N. HACKER ◽  
P. F. LINDEN
Keyword(s):  
Steady State ◽  
Perfect Fluid ◽  
Control Point ◽  
Gravity Current ◽  
Propagation Speed ◽  
Gravity Currents ◽  
Force Balance ◽  
Fluid Approximation ◽  
Left Hand ◽  
Layer Solution

A theory is developed for the speed and structure of steady-state non-dissipative gravity currents in rotating channels. The theory is an extension of that of Benjamin (1968) for non-rotating gravity currents, and in a similar way makes use of the steady-state and perfect-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 non-hydrostatic and ageostrophic motion in a control volume V ahead of the control point, with the solution being determined by the requirements, consistent with the perfect-fluid approximation, of energy and momentum conservation in V, as expressed by Bernoulli's theorem and a generalized flow-force balance. The governing parameter in the problem, which expresses the strength of the background rotation, is the ratio W = B/R, where B is the channel width and R = (g′H)1/2/f is the internal Rossby radius of deformation based on the total depth of the ambient fluid H. Analytic solutions are determined for the particular case of zero front-relative flow within the gravity current. For each value of W there is a unique non-dissipative two-layer solution, and a non-dissipative one-layer solution which is specified by the value of the wall-depth h0. In the two-layer case, the non-dimensional propagation speed c = cf(g′H)−1/2 increases smoothly from the non-rotating value of 0.5 as W increases, asymptoting to unity for W → ∞. The gravity current separates from the left-hand wall of the channel at W = 0.67 and thereafter has decreasing width. The depth of the current at the right-hand wall, h0, increases, reaching the full depth at W = 1.90, after which point the interface outcrops on both the upper and lower boundaries, with the distance over which the interface slopes being 0.881R. In the one-layer case, the wall-depth based propagation speed Froude number c0 = cf(g′h0)−1/2 = 21/2, as in the non-rotating one-layer case. The current separates from the left-hand wall of the channel at W0 ≡ B/R0 = 2−1/2, and thereafter has width 2−1/2R0, where R0 = (g′h0)1/2/f is the wall-depth based deformation radius.

scholarly journals The Sensitivity of Salt Wedge Estuaries to Channel Geometry

2015 ◽  
Vol 45 (12) ◽  
pp. 3169-3183 ◽  
Author(s):  
Anthony R. Poggioli ◽  
Alexander R. Horner-Devine
Keyword(s):  
A Priori ◽  
Control Point ◽  
Numerical Data ◽  
Saline Intrusion ◽  
Hydraulic Control ◽  
Bottom Slope ◽  
Salt Wedge ◽  
Definition Of ◽  
Channel Bottom ◽  
Interfacial Drag

AbstractThe authors develop a two-layer hydraulic model to determine the saline intrusion length in sloped and converging salt wedge estuaries. They find that the nondimensional intrusion length = CiL/hS depends significantly on the channel bottom slope and the rate and magnitude of landward width convergence, in addition to the freshwater Froude number. In the definition of , Ci is a quadratic interfacial drag coefficient, L is the salt wedge intrusion length, and hS is the depth at the mouth of the estuary. Bottom slope is found to limit the saline intrusion length, and this limitation accounts for the deviation of the observed exponent n in a scaling relationship with the river discharge of the form L ~ Q−n from the canonical value of 2 to 2.5 predicted by the theory of Schijf and Schönfeld for a flat, prismatic estuary. The authors find that estuary convergence is important only when the ratio of the slope-limited intrusion length to the convergence length is greater than one, and that the effects of convergence are less significant than those of slope limitation. They compare this model to field and validated numerical data and find that the solution predicts the intrusion length with good accuracy, improving on the flat, prismatic solution by orders of magnitude. While this model has good predictive capability, it is sensitive to Ci and the location of the hydraulic control point, both difficult to determine a priori.

scholarly journals Descent and mixing of the overflow plume from Storfjord in Svalbard: an idealized numerical model study

Ocean Science ◽  
2008 ◽  
Vol 4 (2) ◽  
pp. 115-132 ◽  
Author(s):  
I. Fer ◽  
B. Ådlandsvik
Keyword(s):  
Numerical Model ◽  
Froude Number ◽  
Gravity Current ◽  
Model Performance ◽  
Eddy Diffusivity ◽  
Shelf Break ◽  
Hydraulic Control ◽  
Svalbard Archipelago ◽  
Turbulence Measurements ◽  
Ice Production

Abstract. Storfjorden in the Svalbard Archipelago is a sill-fjord that produces significant volumes of dense, brine-enriched shelf water through ice formation. The dense water produced in the fjord overflows the sill and can reach deep into the Fram Strait. For conditions corresponding to a moderate ice production year, the pathway of the overflow, its descent and evolving water mass properties due to mixing are investigated for the first time using a high resolution 3-D numerical model. An idealized modeling approach forced by a typical annual cycle of buoyancy forcing due to ice production is chosen in a terrain-following vertical co-ordinate. Comparison with observational data, including hydrography, fine resolution current measurements and direct turbulence measurements using a microstructure profiler, gives confidence on the model performance. The model eddy diffusivity profiles contrasted to those inferred from the turbulence measurements give confidence on the skill of the Mellor Yamada scheme in representing sub-grid scale mixing for the Storfjorden overflow, and probably for gravity current modeling, in general. The Storfjorden overflow is characterized by low Froude number dynamics except at the shelf break where the plume narrows, accelerates with speed reaching 0.6 m s−1, yielding local Froude number in excess of unity. The volume flux of the plume increases by five-fold from the sill to downstream of the shelf-break. Rotational hydraulic control is not applicable for transport estimates at the sill using upstream basin information. To the leading order, geostrophy establishes the lateral slope of the plume interface at the sill. This allows for a transport estimate that is consistent with the model results by evaluating a weir relation at the sill.

Self-similar multi-layer exchange flow through a contraction

1996 ◽  
Vol 328 ◽  
pp. 49-66 ◽  
Author(s):  
Anders Engqvist
Keyword(s):  
Control Point ◽  
Flow Regimes ◽  
Wave Mode ◽  
Exchange Flow ◽  
Hydraulic Control ◽  
Coupled Flow ◽  
Density Profiles ◽  
Loosely Coupled ◽  
Self Similar ◽  
Bidirectional Flow

The multi-layer exchange equations for gravitationally driven flows between two basins with stable Boussinesq type of stratification in discrete layers, specified far upstream on either side of a connecting strait, result in a hydraulic control condition that must be satisfied at the narrowest part of the contraction, the control point. If one stagnant layer is present at the control point, the control condition that applies to all layers collectively may be separated into two such conditions that apply independently to two groups of layers going in opposite directions separated by the stagnant layer. Such bidirectional flow regimes exist if the structure of the prespecified density profiles permits each of the opposing groups to vertically reduce their thickness by the ratio 2/3 relative to their upstream thicknesses, leaving space for the stagnant layer to protrude through the contraction. Under these restrictions, the bidirectional flow is controlled by the fastest propagating wave mode and the stationary solution then relies on the superposition of two previously known unidirectional self-similar flow regimes that are completely decoupled. Techniques for their numerical computation are presented. The transition into loosely coupled and fully coupled flow is discussed. The decoupling principle also applies when several non-adjacent stagnant layers are simultaneously present at control in which case multiple groups of decoupled layers flow in alternating directions.

scholarly journals Descent and mixing of the overflow plume from Storfjord in Svalbard: an idealized numerical model study

2007 ◽  
Vol 4 (6) ◽  
pp. 855-895
Author(s):  
I. Fer ◽  
B. Ådlandsvik
Keyword(s):  
Numerical Model ◽  
Froude Number ◽  
Gravity Current ◽  
Model Performance ◽  
Eddy Diffusivity ◽  
Shelf Break ◽  
Hydraulic Control ◽  
Svalbard Archipelago ◽  
Turbulence Measurements ◽  
Ice Production

Abstract. Storfjorden in the Svalbard Archipelago is a sill-fjord that produces significant volumes of dense, brine-enriched shelf water through ice formation. The dense water produced in the fjord overflows the sill and can reach deep into the Fram Strait. For conditions corresponding to a moderate ice production year, the pathway of the overflow, its descent and evolving water mass properties due to mixing are investigated for the first time using a high resolution 3-D numerical model. An idealized modeling approach forced by a typical annual cycle of buoyancy forcing due to ice production is chosen in a terrain-following vertical co-ordinate. Comparison with observational data, including hydrography, fine resolution current measurements and direct turbulence measurements using a microstructure profiler, gives confidence on the model performance. The model eddy diffusivity profiles contrasted to those inferred from the turbulence measurements give confidence on the skill of the Mellor Yamada scheme in representing sub-grid scale mixing for the Storfjorden overflow, and probably for gravity current modeling, in general. The Storfjorden overflow is characterized by low Froude number dynamics except at the shelf break where the plume narrows, accelerates with speed reaching 0.6 m s−1, yielding local Froude number in excess of unity. The volume flux of the plume increases by five-fold from the sill to downstream of the shelf-break. Rotational hydraulic control is not applicable for transport estimates at the sill using upstream basin information. To the leading order, geostrophy establishes the lateral slope of the plume interface at the sill. This allows for a transport estimate that is consistent with the model results by evaluating a weir relation at the sill.

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