A hydrodynamical theory of conservative bounded density currents (original) (raw)
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Hydrodynamics analysis of Density currents
Density Current is formed when a fluid with heavier density than the surrounding fluid flows down an inclined bed. These types of flows are common in nature and can be produced by; salinity, temperature inhomogeneities, or suspended particles of silt and clay. Driven by the density difference between inflow and clear water in reservoirs, density current plunges clear water and moves towards a dam, while density current flows on a sloping bed. The vertical spreading due to water entrainment has an important role in determining the propagation rate in the longitudinal direction. In this work, two-dimensional steady-state salt solutions' density currents were investigated by means of experimental studies and data used in turn to verify the numerical model. In the laboratory experiments, the density current enters the channel via a sluice gate, into a lighter ambient fluid and it moves down-slope. Experiments were performed for different concentrations and discharges. Vertical velocity distributions were measured at various stations by Acoustic Doppler Velocimeter (ADV). Results showed a variety of phenomena depending strongly on the entrance buoyancy flux, and Richardson number. As the discharge increases, maximum velocity and current thickness increase as well, but when concentration decreases, the current thickness increases. In the numerical simulation, the governing equations were solved numerically and k-ω turbulence model was used for closure. The buoyancy term was implemented in the numerical model and its constant was calibrated by experiments. For verification, the height and velocity profiles of the dense layer were compared with the experimental data and a good agreement was found.
Two-dimensional density currents in a confined basin
Geophysical & Astrophysical Fluid Dynamics, 2005
We present new experimental results on the mechanisms through which steady two-dimensional density currents lead to the formation of a stratification in a closed basin. A motivation for this work is to test the underlying assumptions in a diffusive ''filling box'' model that describes the oceanic thermohaline circulation (Hughes, G.O. and Griffiths, R.W., A simple convective model of the global overturning circulation, including effects of entrainment into sinking regions, Ocean Modeling, 2005, submitted.). In particular, they hypothesized that a non-uniform upwelling velocity is due to weak along-slope entrainment in density currents associated with a large horizontal entrainment ratio of E eq $ 0.1. We experimentally measure the relationship between the along-slope entrainment ratio, E, of a density current to the horizontal entrainment ratio, E eq , of an equivalent vertical plume. The along-slope entrainment ratios show the same quantitative decrease with slope as observed by Ellison and Turner (Ellison, T.H. and Turner, J.S., Turbulent entrainment in stratified flows, J. Fluid Mech., 1959, 6, 423-448.), whereas the horizontal entrainment ratio E eq appears to asymptote to a value of E eq ¼ 0:08 at low slopes. Using the measured values of E eq we show that two-dimensional density currents drive circulations that are in good agreement with the two-dimensional filling box model of Baines and Turner (Baines, W.D. and Turner, J.S., Turbulent buoyant convection from a source in a confined region, J. Fluid. Mech., 1969, 37, 51-80.). We find that the vertical velocities of density fronts collapse onto their theoretical prediction that U ¼ À2 À2=3 B 1=3 E 2=3 eq ðH=RÞ, where U is the velocity, H the depth, B the buoyancy flux, R the basin width, E eq the horizontal entrainment ratio and ¼ z=H the dimensionless depth. The density profiles are well fitted with Á ¼ 2 À1=3 B 2=3 E À2=3 eq H À1 ½lnðÞ þ , where is the dimensionless time. Finally, we provide a simple example of a diffusive filling box model, where we show how the density stratification of the deep Caribbean waters (below 1850 m depth) can be described by a balance between a steady two-dimensional entraining density current and vertical diffusion in a triangular basin.
Hydrodynamics analysis of density current using two-equation turbulence k − w model
Density current is formed when fluid heavier than ambient fluid flows down an inclined bed. These flows, which are common phenomena in nature, can be produced by salinity and temperature inhomogeneities, or suspended particles of silt and clay. Driven by density differences between the inflow and clear water in the reservoirs, the density current plunges the clear water and moves toward the dam. While density currents flowing on a sloping bed, the vertical spreading due to water entrainment plays an important role in determining the propagation rate in the longitudinal direction. In this work, two-dimensional density currents were simulated by the w k − turbulent model. A collocated finite volume scheme has been used to simulate the motion of this current which propagates under deep ambient water. The governing equations form an elliptic system of partial differential equations, namely continuity, x-momentum, and y-momentum equations for flow and mass conservation equation for particles and the w k − model equations the for turbulent flow. In this study, density current with uniform velocity and uniform concentration enters a channel via a sluice gate into a lighter ambient fluid and moves forward down-slope. The model has been verified with the experimental data sets. Moreover, results have been compared with the standard ε − k turbulent model and show that the ε − k model has a poor result on this current in comparison with the w k − turbulent model.
Gravity current upstream of a buoyant influx in an open-channel flow: a numerical study
Journal of Fluid Mechanics, 1984
The establishment of an upstream intrusion of a buoyant fluid discharged into an open-channel flow of uniform density and finite depth is studied numerically using the full Navier-Stokes and diffusion equations. The problem is posed as an initialboundary-value problem for the laminar motions of a Boussinesq fluid. The equations are integrated numerically by finite-difference methods. The flow patterns produced are controlled by the influx of buoyancy; therefore they are characterized by an inflow densimetric Froude number. A comparison with available experimental data provides favourable support to the theoretical predictions. The critical value of densimetric Froude number of the source of a vertically downward inflow a t the free surface of a channel is determined. For densimetric Froude number less than critical, an intrusion is established on the upstream side of the source. Because dissipative mechanisms associated with viscosity take a finite time to intervene, the intrusion starts as an inviscid gravity current with a propagation speed greater than the surface velocity of the stream. The front speed is proportional to the phase velocity of long internal waves. Subsequently, the front decelerates as the interfacial friction, and, if applicable, the boundary frictional forces increase simultaneously with mass entrainment across the interface. The current slows down towards a two-zone equilibrium: (1) the zone mcompassing the current behind the frontal zone, where a steady state is approached with respect to the inertial frame of reference; (2) the frontal zone, where the upstream speed approaches a steady speed of frontal advance, albeit small. The upstream intrusion alters the flow pattern of the ambient stream dramatically. A significant feature of both the upstream and downstream currents is the presence of surface convergence with concomitant downwelling near the fronts. As the upstream front decelerates, wavelike disturbances are excited just behind the front a t frequencies characteristic of internal waves. As the front approaches steady state, these disturbances appear to be damped. This problem has practical implications in the design of once-through cooling-water systems for power plants taking their cooling water from rivers.
Density currents at steady regime
Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2010
acceleration of gravity and geometric characteristics of currents along the reservoir. It is possible to notice in the numeric simulations, mainly, the need of complementation of the model that refers to the inclusion of the entrainment coefficient and the analysis in the unsteady regime.
Steady Free-Surface Vortical Flows Parallel to the Horizontal Bottom
The Quarterly Journal of Mechanics and Applied Mathematics, 2011
Steady, free-surface, vortical flows of an inviscid, incompressible, heavy fluid over a horizontal, rigid bottom are considered. All flows of constant depth are described for any Lipschitz vorticity distribution. It is shown that the values of Bernoulli's constant, for which such flows exist, are greater than or equal to some critical value depending on the vorticity. For the critical value, only one flow exists and it is unidirectional. Supercritical flows exist for all values of Bernoulli's constant greater than the critical one; every such flow is also unidirectional and its depth is smaller than that of the critical flow. Furthermore, at least one flow other than supercritical does exist for every value of Bernoulli's constant greater than the critical one. It is found that for some vorticity distributions, the number of constant depth flows increases unrestrictedly as Bernoulli's constant tends to infinity. However, all these flows, except for one or two, have counter-currents; their number depends on Bernoulli's constant and increases by at least two every time when this constant becomes greater than a critical value (the above mentioned is the smallest of them), belonging to a sequence defined by the vorticity. A classification of vorticity distributions is presented; it divides all of them into three classes in accordance with the behaviour of some integral of the distribution on the interval [0, 1]. For distributions in the first class, a unidirectional subcritical flow exists for all admissible values of Bernoulli's constant. For vorticity distributions belonging to the other two classes such a flow exists only when Bernoulli's constant is less than a certain value. If Bernoulli's constant is greater than this value, then at least one flow with counter-currents does exist along with the unidirectional supercritical flow. The second and third classes of vorticity distributions are distinguished from one another by the character of the counter-currents. If a distribution is in the second class, then a near-bottom counter-current is always present for sufficiently large values of Bernoulli's constant. For distributions in the third class, a near-surface counter-current is always present for such values of the constant. Several examples illustrating the results are considered.
Formation of the density currents in the zone of confluence of two rivers
Journal of Hydrology, 2014
The peculiarities of the formation of density currents in the zone of confluence of two rivers with strongly different hydrochemical regimes are studied numerically and experimentally. The three-dimensional numerical simulation shows that the water of the river of higher mineralization and density flows under the water of the river of lower mineralization and density and vice versa. And besides, such overlapping of the water streams is observed both upstream and downstream of the confluence of two rivers. The results of numerical simulation are supported by the data of expedition observations and in situ measurements. A similar phenomenon, namely, a flow of two overlapped oppositely directed water streams was previously discovered in the mouth zone of the rivers flowing in the sea. Our study reveals the existence of a new type of the hydrological systems, in which such a phenomenon occurs.
Numerical and Experimental Study of Flows with Variable Density
2019
The knowledge of density current behaviours as a result of two or more fluids of different densities interacting is of particular importance in many practical applications. Within the field of hydraulic engineering, examples include buoyant effluent discharges from desalination plants, advancements of saline water under freshwater in estuaries, and flows occurring when a gate is removed at the outflow/inflow of a river. The main goal of this study is to improve the understanding of the mixing patterns of density currents as well as their related numerical simulation. In this study, first, an advanced numerical solver for 2D variable-density shallow water equations is developed and validated where both well balanced and positivity preserving properties are achieved over an unstructured grid. The improved numerical scheme is flexible, and accounts for flooding over irregular bed topographies by using a triangular grid. Second, a numerical study of two-layer stratified flows over an is...
2019
The density current is the current caused by imposing gravity on the density difference of two fluids. In case this current enters into another fluid with a lower density, it becomes a subsurface density current. Density currents are generally classified into two groups of permanent stable currents (such as salty density current) and unstable density currents (such as sedimentary current or turbidity currents). The occurrence of this kind of currents in reservoirs of dams leads to the transfer of sediments to the dambody which creates a hazard for the facilities and structures intended to meet the dam's objectives. One of these methods is to remove the sediments of flooding currents through dynamic density current. On the other hand, it is time-consuming and costly to make a physical model for investigating the dynamic density current. Therefore, in this research, investigating the dynamic density current was conductedthrough Flow-3d software. Given that the software has six turbulence of K-e (two-equation), Laminar, Rng, One-equation, Eddy and Prandtl models. To examine the variation of the effect of slope, discharge and concentration on the frontal speed, the body speed profile, as well as the body of water entrainment, was performed for each of the models (for a total of six models, 90 runs) of 15 tests in Flow3d software. Then, the results of the turbulence models were compared to experimental results to obtain a more accurate disturbance model. The results indicated that laminar and eddy models had the most coordination with experimental data, and the laminar disturbance model was preferable to eddy model. Keywords: Density current, Turbulence models, Frontal speed, Body speed profile, Waterentrainment intensity on bod
2D width-averaged numerical simulation of density current in a diverging channel
2011
When a buoyant inflow of higher density enters a reservoir, it sinks below the ambient water and forms an underflow. Downstream of the plunge point, the flow becomes progressively diluted due to the fluid entrainment. This study seeks to explore the ability of 2D width-averaged unsteady Reynolds-averaged Navier-Stokes (RANS) simulation approach for resolving density currents in an inclined diverging channel. A 2D width-averaged unsteady RANS equations closed by a buoyancy-modified ε − k turbulence model are integrated in time with a second-order fractional step approach coupled with a direct implicit method and discretized in space on a staggered mesh using a second-order accurate finite volume approach incorporating a high resolution semi-Lagrangian technique for the convective terms. A series of 2D width-averaged unsteady simulations are carried out for density currents. Comparisons with the experimental measurements and the other numerical simulations show that the predictions of velocity and density field are with reasonable accuracy.