Entraining gravity currents

2013 ◽  
Vol 731 ◽  
pp. 477-508 ◽  
Author(s):  
Christopher G. Johnson ◽  
Andrew J. Hogg
Keyword(s):  
Richardson Number ◽  
Solution Structure ◽  
Large Scale ◽  
Gravity Current ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Water Model ◽  
Experimental Measurements ◽  
Entrainment Rate ◽  
Long Time

AbstractEntrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of the bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as ${t}^{0. 447} $. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as ${t}^{4/ 5} $. In both cases, entrainment is most significant close to the current front.

Axisymmetric, constantly supplied gravity currents at high Reynolds number

2011 ◽  
Vol 675 ◽  
pp. 540-551 ◽  
Author(s):  
ANJA C. SLIM ◽  
HERBERT E. HUPPERT
Keyword(s):  
Reynolds Number ◽  
High Reynolds Number ◽  
Threshold Value ◽  
Gravity Currents ◽  
Water Model ◽  
Closure Condition ◽  
Area Source ◽  
Radial Extent ◽  
Long Time ◽  
High Reynolds

We consider theoretically the long-time evolution of axisymmetric, high Reynolds number, Boussinesq gravity currents supplied by a constant, small-area source of mass and radial momentum in a deep, quiescent ambient. We describe the gravity currents using a shallow-water model with a Froude number closure condition to incorporate ambient form drag at the front and present numerical and asymptotic solutions. The predicted profile consists of an expanding, radially decaying, steady interior that connects via a shock to a deeper, self-similar frontal boundary layer. Controlled by the balance of interior momentum flux and frontal buoyancy across the shock, the front advances as (g′sQ/r1/4s)4/154/5, where g′s is the reduced gravity of the source fluid, Q is the total volume flux, rs is the source radius and is time. A radial momentum source has no effect on this solution below a non-zero threshold value. Above this value, the (virtual) radius over which the flow becomes critical can be used to collapse the solution onto the subthreshold one. We also use a simple parameterization to incorporate the effect of interfacial entrainment, and show that the profile can be substantially modified, although the buoyancy profile and radial extent are less significantly impacted. Our predicted profiles and extents are in reasonable agreement with existing experiments.

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.

Compressible particle-driven gravity currents

2001 ◽  
Vol 445 ◽  
pp. 305-325 ◽  
Author(s):  
MARY-LOUISE E. TIMMERMANS ◽  
JOHN R. LISTER ◽  
HERBERT E. HUPPERT
Keyword(s):  
Large Scale ◽  
Numerical Solutions ◽  
Volcanic Eruptions ◽  
Gravity Currents ◽  
Scale Height ◽  
Pyroclastic Flows ◽  
Water Model ◽  
Model Approximation ◽  
Particle Settling ◽  
Density Scale

Large-scale particle-driven gravity currents occur in the atmosphere, often in the form of pyroclastic flows that result from explosive volcanic eruptions. The behaviour of these gravity currents is analysed here and it is shown that compressibility can be important in flow of such particle-laden gases because the presence of particles greatly reduces the density scale height, so that variations in density due to compressibility are significant over the thickness of the flow. A shallow-water model of the flow is developed, which incorporates the contribution of particles to the density and thermodynamics of the flow. Analytical similarity solutions and numerical solutions of the model equations are derived. The gas–particle mixture decompresses upon gravitational collapse and such flows have faster propagation speeds than incompressible currents of the same dimensions. Once a compressible current has spread sufficiently that its thickness is less than the density scale height it can be treated as incompressible. A simple ‘box-model’ approximation is developed to determine the effects of particle settling. The major effect is that a small amount of particle settling increases the density scale height of the particle-laden mixture and leads to a more rapid decompression of the current.

scholarly journals Stratified gravity currents in porous media

2016 ◽  
Vol 791 ◽  
pp. 329-357 ◽  
Author(s):  
Samuel S. Pegler ◽  
Herbert E. Huppert ◽  
Jerome A. Neufeld
Keyword(s):  
Porous Media ◽  
Density Distribution ◽  
Vertical Structure ◽  
Gravity Current ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Density Field ◽  
Vertical Position ◽  
Uniform Density ◽  
Convex Shape

We consider theoretically and experimentally the propagation in porous media of variable-density gravity currents containing a stably stratified density field, with most previous studies of gravity currents having focused on cases of uniform density. New thin-layer equations are developed to describe stably stratified fluid flows in which the density field is materially advected with the flow. Similarity solutions describing both the fixed-volume release of a distributed density stratification and the continuous input of fluid containing a distribution of densities are obtained. The results indicate that the density distribution of the stratification significantly influences the vertical structure of the gravity current. When more mass is distributed into lighter densities, it is found that the shape of the current changes from the convex shape familiar from studies of the uniform-density case to a concave shape in which lighter fluid accumulates primarily vertically above the origin of the current. For a constant-volume release, the density contours stratify horizontally, a simplification which is used to develop analytical solutions. For currents introduced continuously, the horizontal velocity varies with vertical position, a feature which does not apply to uniform-density gravity currents in porous media. Despite significant effects on vertical structure, the density distribution has almost no effect on overall horizontal propagation, for a given total mass. Good agreement with data from a laboratory study confirms the predictions of the model.

A model for the spreading and compaction of two-phase viscous gravity currents

2009 ◽  
Vol 630 ◽  
pp. 299-329 ◽  
Author(s):  
CHLOÉ MICHAUT ◽  
DAVID BERCOVICI
Keyword(s):  
Large Scale ◽  
Gravity Current ◽  
Fluid Phase ◽  
Gravity Currents ◽  
Small Scale ◽  
Fluid Loss ◽  
Large Area ◽  
Finite Radius ◽  
Two Phase ◽  
Mixture Density

Two-phase viscous gravity current theory has numerous applications in the natural sciences, from small-scale lava, sedimentary and glacial flows, to large-scale flows of partially molten mantle. We develop the general equations for two-phase viscous gravity currents composed of a high viscosity matrix and low viscosity fluid for both constant volume and constant flux conditions. A loss of fluid phase is taken into account at the current's upper boundary and corresponds to the degassing of a lava flow or loss of water in sedimentary flows. As the current spreads, its surface increases and fluid loss is facilitated, which modifies the mixture density and viscosity and thus the current's shape; hence spreading of the flow affects fluid loss and vice-versa. Our results show that two-phase gravity currents retain and transport the fluid out to large distances, but the fluid is almost entirely lost within a region of finite radius. This ‘loss radius’ depends on the flow's volume or flux, fluid and matrix properties as well as on the size of fluid parcels or matrix permeability. Application to lava flows shows that degassing occurs over a large area, which affects gas release and transport in the atmosphere.

Gravity currents from a line source in an ambient flow

2008 ◽  
Vol 606 ◽  
pp. 1-26 ◽  
Author(s):  
ANJA C. SLIM ◽  
HERBERT E. HUPPERT
Keyword(s):  
Line Source ◽  
Source Region ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Flow Speed ◽  
Moving Frame ◽  
Ambient Flow ◽  
Constant Rate ◽  
Long Time ◽  
High Reynolds

We present a mainly theoretical study of high-Reynolds-number planar gravity currents in a uniformly flowing deep ambient. The gravity currents are generated by a constant line source of fluid, and may also be supplied with a source of horizontal momentum and a source of particles. We model the motion using a shallow-water approximation and represent the effects of the ambient flow by imposing a Froude-number condition in a moving frame. We present analytic and numerical expressions for the threshold ambient flow speed above which no upstream propagation can occur at long times. For homogeneous gravity currents in an ambient flow below threshold, we find similarity solutions in which the up- and downstream fronts spread at a constant rate and the current propagates indefinitely in both directions. For gravity currents consisting of both interstitial fluid of a different density to the ambient and a sedimenting particle load, we find long-time asymptotic solutions for ambient flow strengths below threshold. These consist of a steady particle-rich near-source region, in which settling and advection of particles balance, and an effectively particle-free frontal region. The homogeneous behaviour of the fronts ensures that they also spread at a constant rate and therefore can propagate upstream indefinitely. For gravity currents driven solely by a sedimenting particle load, we find numerically that a single regime exists for ambient flow strengths below threshold. In these solutions, settling balances advection near the source leading to a steady region, which joins on to a complex frontal boundary layer. The upstream front progressively decelerates. Our solutions for homogeneous and particle-driven gravity currents compare well with published experimental results.

scholarly journals Estimation of friction parameters in gravity currents by data assimilation in a model hierarchy

2011 ◽  
Vol 8 (1) ◽  
pp. 159-187 ◽  
Author(s):  
A. Wirth
Keyword(s):  
Data Assimilation ◽  
Gravity Current ◽  
Gravity Currents ◽  
Water Model ◽  
Friction Laws ◽  
Quadratic Drag ◽  
Small Reynolds Numbers ◽  
Model Hierarchy ◽  
Using Data ◽  
Friction Parameters

Abstract. This paper is the last in a series of three investigating the friction laws and their parametrisation in idealised gravity currents in a rotating frame. Results on the dynamics of a gravity current (Wirth, 2009) and on the estimation of friction laws by data assimilation (Wirth and Verron, 2008) are combined to estimate the friction parameters and discriminate between friction laws in non-hydrostatic numerical simulations of gravity current dynamics, using data assimilation and a reduced gravity shallow water model. I demonstrate, that friction parameters and laws in gravity currents can be estimated using data assimilation. The results clearly show that friction follows a linear Rayleigh law for small Reynolds numbers and the estimated value agrees well with the analytical value obtained for non-accelerating Ekman layers. A significant and sudden departure towards a quadratic drag law at an Ekman layer based Reynolds number of around 800 is shown, in agreement with classical laboratory experiments. The drag coefficient obtained compare well to friction values over smooth surfaces. I show that data assimilation can be used to determine friction parameters and discriminate between friction laws and that it is a powerful tool in systematically connection models within a model hierarchy.

The effect of confining impermeable boundaries on gravity currents in a porous medium

2010 ◽  
Vol 649 ◽  
pp. 1-17 ◽  
Author(s):  
MADELEINE J. GOLDING ◽  
HERBERT E. HUPPERT
Keyword(s):  
Free Surface ◽  
Gravity Current ◽  
Mass Conservation ◽  
Propagation Rate ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Constant Flux ◽  
Channel Shape ◽  
Dimensional Gravity ◽  
Uniform Cross Section

The effect of confining boundaries on gravity currents in porous media is investigated theoretically and experimentally. Similarity solutions are derived for currents when the volume increases as tα in horizontal channels of uniform cross-section with boundary height b satisfying b ~ a|y/a|n, where y is the cross-channel coordinate and a is a length scale of the channel width. Experiments were carried out in V-shaped and semicircular channels for the case of gravity currents with constant volume (α=0) and constant flux (α=1). These showed generally good agreement with the theory.Typically, we find that the propagation of the current is well described by L ~ tc for some scalar c. We study the dependence of c on the time exponent of the volume of fluid in the current, α, and the geometry of the channel, parameterized by n. For all channel shapes, there exists a critical value of α, αc = 1/2, above which increasing n causes an increase in c and below which increasing n causes a decrease in c, where increasing n corresponds to opening up the channel boundary to the horizontal. The current height increases or decreases with respect to time depending on whether α is greater or less than αc. It is this fact, along with global mass conservation, which explains why varying the channel shape n affects the propagation rate c in different ways depending on α.We also consider channels inclined at an angle θ to the horizontal. When the slope of the channel is much greater than the slope of the free surface of the current, the component of gravity parallel to the slope dominates, causing the current to move with a constant velocity, Vf say, regardless of channel shape n and flux parameter α, in agreement with results for a two-dimensional gravity current obtained by Huppert & Woods (1995) and some initially axisymmetric gravity currents presented by Vella & Huppert (2006). If the effect of the component of gravity perpendicular to the channel may not be neglected, i.e. if the slopes of the channel and free surface of the current are comparable, we find that, in a frame moving with speed Vf, the form of the governing equation for the height of a current in an equivalent horizontal channel is recovered. We calculate that the height of a constant flux gravity current down an inclined channel will tend to a fixed depth, which is determined by the channel shape, n, and the physical properties of the fluid and rock. Experimental and numerical results for inclined V-shaped channels agree very well with this theory.

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.

Particle-driven gravity currents down planar slopes

1999 ◽  
Vol 390 ◽  
pp. 75-91 ◽  
Author(s):  
ROGER T. BONNECAZE ◽  
JOHN R. LISTER
Keyword(s):  
Gravity Current ◽  
Gravity Currents ◽  
Similarity Solutions ◽  
Bottom Friction ◽  
Turbidity Currents ◽  
Scaling Analysis ◽  
Ambient Fluid ◽  
Buoyancy Driven Flows ◽  
Depositional Patterns ◽  
Planar Slope

Particle-driven gravity currents, as exemplified by either turbidity currents in the ocean or ignimbrite flows in the atmosphere, are buoyancy-driven flows due to a suspension of dense particles in an ambient fluid. We present a theoretical study on the dynamics of and deposition from a turbulent current flowing down a uniform planar slope from a constant-flux point source of particle-laden fluid. The flow is modelled using the shallow-water equations, including the effects of bottom friction and entrainment of ambient fluid, coupled to an equation for the transport and settling of the particles. Two flow regimes are identified. Near the source and for mild slopes, the flow is dominated by a balance between buoyancy and bottom friction. Further downstream and for steeper slopes, entrainment also affects the behaviour of the current. Similarity solutions are also developed for the simple cases of homogeneous gravity currents with no settling of particles in the friction-dominated and entrainment-dominated regimes. Estimates of the width and length of the deposit from a monodisperse particle-driven gravity current with settling are derived from scaling analysis for each regime, and the contours of the depositional patterns are determined from numerical solution of the governing equations.

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