scholarly journals Mixing in axisymmetric gravity currents

2015 ◽  
Vol 782 ◽  
Author(s):  
Peeradon Samasiri ◽  
Andrew W. Woods
Keyword(s):  
Light Attenuation ◽  
Gravity Current ◽  
Finite Depth ◽  
Aqueous Salt Solution ◽  
Vertical Shear ◽  
Ambient Fluid ◽  
Averaged Equations ◽  
Self Similar Solution ◽  
Diverging Channel ◽  
High Reynolds

We present new experiments to measure the rate of entrainment of ambient fluid into a high Reynolds number, axisymmetric, turbulent gravity current. The current is produced by the rapid release of a finite volume of aqueous salt solution from a lock of length $r_{o}$ into a diverging channel, $r>0$, of angle $9.5^{\circ }$, filled with a finite depth, $H$, of fresh water. Using light attenuation we measure the evolving density of the flow, and using dye studies we illustrate the process of mixing between the current and ambient fluid. After an initial adjustment, a circulation develops in the head of the flow: current fluid reaches the nose of the flow, rises up and moves backwards relative to the nose. We find that, owing to the mixing, the volume of the current increases as $V\sim 0.2r_{n}^{7/4}r_{o}^{1/4}H$ while the maximum depth of the head decreases as $h_{n}\sim 0.5H(r_{o}/r_{n})^{1/4}$, where $r_{n}$ is the location of the front of the current. Combining these results, we estimate that the recirculating current fluid mixes with a fraction $E=0.33\pm 0.09$ of the ambient fluid that is directly ahead of the current and displaced upwards by it. Some of the mixed fluid supplies the tail of the flow, while the remainder recirculates into the head, which becomes progressively more dilute. In accord with Huppert & Simpson (J. Fluid Mech., vol. 99, 1980, pp. 785–799), we find that the position of the front increases with time as $r_{n}\approx (1.28\pm 0.05)B^{1/4}t^{1/2}$, where $B$ is the total buoyancy of the flow. We also find that the maximum value of the vertical integral of the buoyancy $(\overline{g^{\prime }}h)_{n}$ decreases with the position of the nose according to the relation $(\overline{g^{\prime }}h)_{n}\approx (0.89\pm 0.12)Br_{n}^{-2}$, consistent with a Froude number $0.86\pm 0.07$. We compare our measurements with a new idealised self-similar solution of the depth-averaged equations that accounts for the mixing at the nose, the vertical shear in the velocity and the lateral stratification of the buoyancy within the current.

Gravity currents: entrainment, stratification and self-similarity

2015 ◽  
Vol 784 ◽  
pp. 130-162 ◽  
Author(s):  
Diana Sher ◽  
Andrew W. Woods
Keyword(s):  
Light Attenuation ◽  
Gravity Current ◽  
Finite Depth ◽  
Reduced Gravity ◽  
Ambient Fluid ◽  
Tail Region ◽  
Self Similar Solution ◽  
Self Similar ◽  
Two Phases ◽  
Lock Gate

We present new experiments of the motion of a turbulent gravity current produced by the rapid release of a finite volume of dense aqueous solution from a lock of length $L$ into a channel $x>0$ filled with a finite depth, $H$, of fresh water. Using light attenuation we measure the mixing and evolving density of the flow, and, using dye studies, we follow the motion of the current and the ambient fluid. After the fluid has slumped to the base of the tank, there are two phases of the flow. When the front of the current, $x_{n}$, is within the region $2L<x_{n}<7L$, the fluid in the head of the current retains its original density and the flow travels with a constant speed. We find that approximately $0.75(\pm 0.05)$ of the ambient fluid displaced by the head mixes with the fluid in the head. The mixture rises over the head and feeds a growing stratified tail region of the flow. Dye studies show that fluid with the original density continues to reach the front of the current, at a speed which we estimate to be approximately $1.35\pm 0.05$ times that of the front, consistent with data of Berson (Q. J. R. Meteorol. Soc., vol. 84, 1958, pp. 1–16) and Kneller et al. (J. Geophys. Res. Oceans, vol. 104, 1999, pp. 5281–5291). This speed is similar to that of the ‘bore’, the trailing edge of the original lock gate fluid, as described by Rottman & Simpson (J. Fluid Mech., vol. 135, 1983, pp. 95–110). The continual mixing at the front leads to a gradual decrease of the mass of unmixed original lock gate fluid. Eventually, when the nose extends beyond $x_{n}\approx 7L$, the majority of the lock gate fluid has been diluted through the mixing. As the current continues, it adjusts to a second regime in which the position of the head increases with time as $x_{n}\approx 1.7B^{1/3}t^{2/3}$, where $B$ is the total buoyancy of the flow per unit width across the channel, while the depth-averaged reduced gravity in the head decreases through mixing with the ambient fluid according to the relation $g_{n}^{\prime }\approx 4.6H^{-1}B^{2/3}t^{-2/3}$. Measurements also show that the depth of the head $h_{n}(t)$ is approximately constant, $h_{n}\sim 0.38H$, and the reduced gravity decreases with height above the base of the current and with distance behind the front of the flow. Using the depth-averaged shallow-water equations, we derive a new class of self-similar solution which models the lateral structure of the flow by assuming the ambient fluid is entrained into the current in the head of the flow. By comparison with our data, we estimate that a fraction $0.69\pm 0.06$ of the ambient fluid displaced by the head of the current is mixed into the flow in this approximately self-similar regime, and the front of the current has a Froude number $0.9\pm 0.05$. We discuss the implications of our results for the evolution of the buoyancy in a gravity current as a function of the distance from the source.

Mixing in continuous gravity currents

2017 ◽  
Vol 818 ◽  
Author(s):  
Diana Sher ◽  
Andrew W. Woods
Keyword(s):  
Light Attenuation ◽  
Vertical Gradient ◽  
Gravity Currents ◽  
Mixing Zone ◽  
Ambient Fluid ◽  
Unit Width ◽  
A Value ◽  
Dense Fluid ◽  
Catch Up ◽  
High Reynolds

We present measurements of the entrainment of ambient fluid into high-Reynolds-number gravity currents produced by a steady flux of buoyancy. The currents propagate along a horizontal channel and the mixing is measured using a light attenuation technique to obtain the cross-channel average of the density throughout the current. The total volume of the current increases linearly with time, at a rate in the range $(1.8{-}2.1)Q_{o}$ for source Froude numbers, $Fr_{o}$, in the range $0.1{-}3.7$, where $Q_{o}$ is the source volume flux per unit width. Most mixing occurs either immediately downstream of the inflow or near the head of the flow, with an increasing proportion of the entrainment occurring in a mixing zone near the inflow as $Fr_{o}$ increases. A vertical gradient in the density and horizontal velocity develops in this mixing zone. This enables relatively dense fluid at the base of the current to catch up with the head, where it rises and mixes with the ambient fluid which is displaced over the head. The mixed fluid continues forward more slowly than the head, forming the relatively dilute fluid in the upper part of the current. Our data show that the depth and the depth-averaged buoyancy are primarily dependent on the position relative to the front, with the speed of the front being $\unicode[STIX]{x1D706}(Fr_{o})B_{o}^{1/3}$, where $B_{o}$ is the source buoyancy flux per unit width. Here, $\unicode[STIX]{x1D706}(Fr_{o})$ increases from 0.9 to 1.1 as $Fr_{o}$ increases from 0.1 to 3.7, while the Froude number at the head of the flow has a value of $1.1\pm 0.05$.

Inertial gravity currents produced by fluid drainage from an edge

2017 ◽  
Vol 827 ◽  
pp. 640-663 ◽  
Author(s):  
Mostafa Momen ◽  
Zhong Zheng ◽  
Elie Bou-Zeid ◽  
Howard A. Stone
Keyword(s):  
Differential Equations ◽  
Gravity Current ◽  
Experimental Studies ◽  
Dynamical Behaviour ◽  
Rectangular Domain ◽  
Dimensionless Time ◽  
Horizontal Length ◽  
High Reynolds ◽  
End Wall ◽  
Similar Phase

We present theoretical, numerical and experimental studies of the release of a finite volume of fluid instantaneously from an edge of a rectangular domain for high Reynolds number flows. For the cases we considered, the results indicate that approximately half of the initial volume exits during an early adjustment period. Then, the inertial gravity current reaches a self-similar phase during which approximately 40 % of its volume drains and its height decreases as $\unicode[STIX]{x1D70F}^{-2}$, where $\unicode[STIX]{x1D70F}$ is a dimensionless time that is derived with the typical gravity wave speed and the horizontal length of the domain. Based on scaling arguments, we reduce the shallow-water partial differential equations into two nonlinear ordinary differential equations (representing the continuity and momentum equations), which are solved analytically by imposing a zero velocity boundary condition at the closed end wall and a critical Froude number condition at the open edge. The solutions are in good agreement with the performed experiments and direct numerical simulations for various geometries, densities and viscosities. This study provides new insights into the dynamical behaviour of a fluid draining from an edge in the inertial regime. The solutions may be useful for environmental, geophysical and engineering applications such as open channel flows, ventilations and dam-break problems.

On symmetric intrusions in a linearly stratified ambient: a revisit of Benjamin's steady-state propagation results

2021 ◽  
Vol 929 ◽  
Author(s):  
M. Ungarish
Keyword(s):  
Steady State ◽  
Control Volume ◽  
Gravity Current ◽  
Head Loss ◽  
Stability Criteria ◽  
Ambient Fluid ◽  
Height Ratio ◽  
Stagnation Line ◽  
Volume Analysis ◽  
State Propagation

Previous studies have extended Benjamin's theory for an inertial steady-state gravity current of density $\rho _{c}$ in a homogeneous ambient fluid of density $\rho _{o} < \rho _{c}$ to the counterpart propagation in a linearly stratified (Boussinesq) ambient (density decreases from $\rho _b$ to $\rho _{o}$ ). The extension is typified by the parameter $S = (\rho _{b}-\rho _{o})/(\rho _{c}-\rho _{o}) \in (0,1]$ , uses Long's solution for the flow over a topography to model the flow of the ambient over the gravity current, and reduces well to the classical theory for small and moderate values of $S$ . However, for $S=1$ , i.e. $\rho _b = \rho _c$ , which corresponds to a symmetric intrusion, various idiosyncrasies appear. Here attention is focused on this case. The control-volume analysis (balance of volume, mass, momentum and vorticity) produces a fairly compact analytical formulation, pending a closure for the head loss, and subject to stability criteria (no inverse stratification downstream). However, we show that plausible closures that work well for the non-stratified current (like zero head loss on the stagnation line, or zero vorticity diffusion) do not produce satisfactory results for the intrusion (except for some small ranges of the height ratio of current to channel, $a = h/H$ ). The reasons and insights are discussed. Accurate data needed for comparison with the theoretical model are scarce, and a message of this paper is that dedicated experiments and simulations are needed for the clarification and improvement of the theory.

scholarly journals An experimental and theoretical study of the dynamics of grounding lines

2013 ◽  
Vol 728 ◽  
pp. 5-28 ◽  
Author(s):  
Samuel S. Pegler ◽  
M. Grae Worster
Keyword(s):  
Theoretical Study ◽  
Fluid Layer ◽  
Gravity Current ◽  
Three Dimensional ◽  
Vertical Shear ◽  
Inviscid Fluid ◽  
West Antarctica ◽  
Full Contact ◽  
Grounding Line ◽  
Longitudinal Extension

AbstractWe present an experimental and theoretical study of a thin, viscous fluid layer that flows radially under gravity from a point source into a denser inviscid fluid layer of uniform depth above a rigid horizontal surface. Near the source, the viscous layer lies in full contact with the surface, forming a vertical-shear-dominated viscous gravity current. At a certain distance from the source, the layer detaches from the surface to form a floating current whose dynamics are controlled by the viscous stresses due to longitudinal extension. We describe the dynamics of the grounded and floating components using distinct thin-layer theories. Separating the grounded and floating regions is the freely moving line of detachment, or grounding line, whose evolution we model by balancing the horizontal forces between the two regions. Using numerical and asymptotic analysis, we calculate the evolution of the system from a self-similar form at early times towards a steady state at late times. We use our solutions to illustrate how three-dimensional stresses within marine ice sheets, such as that of West Antarctica, can lead to stabilization of the grounding line. To assess the validity of the assumptions underlying our model, we compare its predictions with data from a series of laboratory experiments.

Self‐similar solution of the second kind for a convergent viscous gravity current

1992 ◽  
Vol 4 (6) ◽  
pp. 1148-1155 ◽  
Author(s):  
Javier Alberto Diez ◽  
Roberto Gratton ◽  
Julio Gratton
Keyword(s):  
Gravity Current ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Self Similar

Energy balances for axisymmetric gravity currents in homogeneous and linearly stratified ambients

2008 ◽  
Vol 616 ◽  
pp. 303-326 ◽  
Author(s):  
MARIUS UNGARISH ◽  
HERBERT E. HUPPERT
Keyword(s):  
High Reynolds Number ◽  
Gravity Current ◽  
Water Model ◽  
Navier Stokes ◽  
Shallow Water Model ◽  
Fluid System ◽  
Dense Fluid ◽  
High Reynolds ◽  
Energy Balances ◽  
Two Fluid

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.

On the slow draining of a gravity current moving through a layered permeable medium

2001 ◽  
Vol 444 ◽  
pp. 23-47 ◽  
Author(s):  
DAVID PRITCHARD ◽  
ANDREW W. WOODS ◽  
ANDREW J. HOGG
Keyword(s):  
Initial Conditions ◽  
Gravity Current ◽  
Numerical Data ◽  
Governing Equations ◽  
Dense Fluid ◽  
Self Similar Solution ◽  
Lower Permeability ◽  
Finite Distance ◽  
Self Similar ◽  
Initial Source

We examine the gravitational dispersal of dense fluid through a horizontal permeable layer, which is separated from a second underlying layer by a narrow band of much lower permeability. We derive a series of analytical solutions which describe the propagation of the fluid through the upper layer and the draining of the fluid into the underlying region. The model predicts that the current initially spreads according to a self-similar solution. However, as the drainage becomes established, the spreading slows, and in fact the fluid only spreads a finite distance before it has fully drained into the underlying layer. We examine the sensitivity of the results to the initial conditions through numerical solution of the governing equations. We find that for sources of sufficiently large initial aspect ratio (defined as the ratio of height to length), the solution converges rapidly to the initially self-similar regime. For longer and shallower initial source conditions, this convergence does not occur, but we derive estimates for the run-out length of the current, which compare favourably with our numerical data. We also present some preliminary laboratory experiments, which support the model.

Inclined gravity currents filling basins: the impact of peeling detrainment on transport and vertical structure

2017 ◽  
Vol 820 ◽  
pp. 400-423 ◽  
Author(s):  
Charlie A. R. Hogg ◽  
Stuart B. Dalziel ◽  
Herbert E. Huppert ◽  
Jörg Imberger
Keyword(s):  
Gravity Current ◽  
Uniform Density ◽  
Ambient Fluid ◽  
Transport Pathway ◽  
Density Profiles ◽  
Model Match ◽  
Dense Fluid ◽  
Vertical Density ◽  
Multiprocess Models ◽  
The Impact

Transport of dense fluid by an inclined gravity current can control the vertical density structure of the receiving basin in many natural and industrial settings. A case familiar to many is a lake fed by river water that is dense relative to the lake water. In laboratory experiments, we pulsed dye into the basin inflow to visualise the transport pathway of the inflow fluid through the basin. We also measured the evolving density profile as the basin filled. The experiments confirmed previous observations that when the turbulent gravity current travelled through ambient fluid of uniform density, only entrainment into the dense current occurred. When the gravity current travelled through the stratified part of the ambient fluid, however, the outer layers of the gravity current outflowed from the current by a peeling detrainment mechanism and moved directly into the ambient fluid over a large range of depths. The prevailing model of a filling box flow assumes that a persistently entraining gravity current entrains fluid from the basin as the current descends to the deepest point in the basin. This model, however, is inconsistent with the transport pathway observed in visualisations and poorly matches the stratifications measured in basin experiments. The main contribution of the present work is to extend the prevailing filling box model by incorporating the observed peeling detrainment. The analytical expressions given by the peeling detrainment model match the experimental observations of the density profiles more closely than the persistently entraining model. Incorporating peeling detrainment into multiprocess models of geophysical systems, such as lakes, will lead to models that better describe inflow behaviour.

Fountains impinging on a density interface

2008 ◽  
Vol 595 ◽  
pp. 115-139 ◽  
Author(s):  
JOSEPH K. ANSONG ◽  
PATRICK J. KYBA ◽  
BRUCE R. SUTHERLAND
Keyword(s):  
Experimental Study ◽  
Relative Density ◽  
Constant Velocity ◽  
Gravity Current ◽  
Return Flow ◽  
Experimental Measurements ◽  
Ambient Fluid ◽  
Initial Momentum ◽  
Front Position ◽  
Homogeneous Environment

We present an experimental study of an axisymmetric turbulent fountain in a two-layer stratified environment. Interacting with the interface, the fountain is observed to exhibit three regimes of flow. It may penetrate the interface, but nonetheless return to the source where it spreads as a radially propagating gravity current; the return flow may be trapped at the interface where it spreads as a radially propagating intrusion or it may do both. These regimes have been classified using empirically determined regime parameters which govern the relative initial momentum of the fountain and the relative density difference of the fountain and the ambient fluid. The maximum vertical distance travelled by the fountain in a two-layer fluid has been theoretically determined by extending the theory developed for fountains in a homogeneous environment. The theory compares favourably with experimental measurements. We have also developed a theory to analyse the initial speeds of the resulting radial currents. The spreading currents exhibited two different flow regimes: a constant-velocity regime and an inertia-buoyancy regime in which the front position, R, scales with time, t, as R ∼ t3/4. These regimes were classified using a critical Froude number which characterized the competing effects of momentum and buoyancy in the currents.

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