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.

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.

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.

scholarly journals Particle-driven gravity currents

1993 ◽  
Vol 250 ◽  
pp. 339-369 ◽  
Author(s):  
Roger T. Bonnecaze ◽  
Herbert E. Huppert ◽  
John R. Lister
Keyword(s):  
Particle Concentration ◽  
Local Density ◽  
Gravity Current ◽  
Pressure Head ◽  
Viscous Sublayer ◽  
Gravity Currents ◽  
Momentum Balance ◽  
Exchange Flow ◽  
Mean Flow ◽  
Ambient Fluid

Gravity currents created by the release of a fixed volume of a suspension into a lighter ambient fluid are studied theoretically and experimentally. The greater density of the current and the buoyancy force driving its motion arise primarily from dense particles suspended in the interstitial fluid of the current. The dynamics of the current are assumed to be dominated by a balance between inertial and buoyancy forces; viscous forces are assumed negligible. The currents considered are two-dimensional and flow over a rigid horizontal surface. The flow is modelled by either the single- or the two-layer shallow-water equations, the two-layer equations being necessary to include the effects of the overlying fluid, which are important when the depth of the current is comparable to the depth of the overlying fluid. Because the local density of the gravity current depends on the concentration of particles, the buoyancy contribution to the momentum balance depends on the variation of the particle concentration. A transport equation for the particle concentration is derived by assuming that the particles are vertically well-mixed by the turbulence in the current, are advected by the mean flow and settle out through the viscous sublayer at the bottom of the current. The boundary condition at the moving front of the current relates the velocity and the pressure head at that point. The resulting equations are solved numerically, which reveals that two types of shock can occur in the current. In the late stages of all particle-driven gravity currents, an internal bore develops that separates a particle-free jet-like flow in the rear from a dense gravity-current flow near the front. The second type of bore occurs if the initial height of the current is comparable to the depth of the ambient fluid. This bore develops during the early lock-exchange flow between the two fluids and strongly changes the structure of the current and its transport of particles from those of a current in very deep surroundings. To test the theory, several experiments were performed to measure the length of particle-driven gravity currents as a function of time and their deposition patterns for a variety of particle sizes and initial masses of sediment. The comparison between the theoretical predictions, which have no adjustable parameters, and the experimental results are very good.

Critical regime of gravity currents flowing in non-rectangular channels with density stratification

2018 ◽  
Vol 840 ◽  
pp. 579-612
Author(s):  
L. Chiapponi ◽  
M. Ungarish ◽  
S. Longo ◽  
V. Di Federico ◽  
F. Addona
Keyword(s):  
Cross Section ◽  
Power Law ◽  
Rectangular Cross Section ◽  
Gravity Current ◽  
Gravity Currents ◽  
Critical Conditions ◽  
Uniform Density ◽  
Ambient Fluid ◽  
Real Numbers ◽  
Positive Real

We present theoretical and experimental analyses of the critical condition where the inertial–buoyancy or viscous–buoyancy regime is preserved in a uniform-density gravity current (which propagates over a horizontal plane) of time-variable volume ${\mathcal{V}}=qt^{\unicode[STIX]{x1D6FF}}$ in a power-law cross-section (with width described by $f(z)=bz^{\unicode[STIX]{x1D6FC}}$, where $z$ is the vertical coordinate, $b$ and $q$ are positive real numbers, and $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$ are non-negative real numbers) occupied by homogeneous or linearly stratified ambient fluid. The magnitude of the ambient stratification is represented by the parameter $S$, with $S=0$ and $S=1$ describing the homogeneous and maximum stratification cases respectively. Earlier theoretical and experimental results valid for a rectangular cross-section ($\unicode[STIX]{x1D6FC}=0$) and uniform ambient fluid are generalized here to a power-law cross-section and stratified ambient. Novel time scalings, obtained for inertial and viscous regimes, allow a derivation of the critical flow parameter $\unicode[STIX]{x1D6FF}_{c}$ and the corresponding propagation rate as $Kt^{\unicode[STIX]{x1D6FD}_{c}}$ as a function of the problem parameters. Estimates of the transition length between the inertial and viscous regimes are also derived. A series of experiments conducted in a semicircular cross-section ($\unicode[STIX]{x1D6FC}=1/2$) validate the critical values $\unicode[STIX]{x1D6FF}_{c}=2$ and $\unicode[STIX]{x1D6FF}_{c}=9/4$ for the two cases $S=0$ and $1$. The ratio between the inertial and viscous forces is determined by an effective Reynolds number proportional to $q$ at some power. The threshold value of this number, which enables a determination of the regime of the current (inertial–buoyancy or viscous–buoyancy) in critical conditions, is determined experimentally for both $S=0$ and $S=1$. We conclude that a very significant generalization of the insights and results from two-dimensional (rectangular cross-section channel) gravity currents to power-law cross-sections is available.

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.

On gravity currents propagating at the base of a stratified ambient

2002 ◽  
Vol 458 ◽  
pp. 283-301 ◽  
Author(s):  
MARIUS UNGARISH ◽  
HERBERT E. HUPPERT
Keyword(s):  
Shallow Water ◽  
Stokes Equations ◽  
Gravity Current ◽  
Gravity Currents ◽  
Navier Stokes ◽  
Theoretical Interpretation ◽  
Ambient Fluid ◽  
Navier Stokes Equations ◽  
Speed Of Propagation ◽  
Good Agreement

The behaviour of an inviscid gravity current which is released from behind a lock and then propagates over a horizontal boundary at the base of a stratified ambient fluid is considered. An extension of the shallow-water formulation for a homogeneous ambient to the stratified case is developed, without using any additional adjustable parameters. Attention is focused on the initial ‘slumping’ stage of a rectangular current which is typified by a constant speed of propagation. The analytical results are in good agreement with, and give a firm theoretical interpretation of, the corresponding experiments and numerical simulations of Maxworthy et al. (2002). Finite-difference solutions of the Navier–Stokes equations, using a different technique from that used by Maxworthy et al. (2002), are also presented and provide both good agreement with their results and further validation of the present shallow-water approach. The differences between currents in a homogeneous and stratified ambient, and possible implementation of the results to other configurations, are discussed.

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.

A flow-front instability in viscous gravity currents

1998 ◽  
Vol 369 ◽  
pp. 1-21 ◽  
Author(s):  
DON SNYDER ◽  
STEPHEN TAIT
Keyword(s):  
Gravitational Instability ◽  
Gravity Current ◽  
High Viscosity ◽  
Gravity Currents ◽  
Ambient Fluid ◽  
Cell Width ◽  
Dense Fluid ◽  
Unstable Layer ◽  
Gravitational Mechanism ◽  
Hele Shaw Cell

We describe an instability that appears at the front of laminar gravity currents as they intrude into a viscous, miscible ambient fluid. The instability causes a current to segment into fingers aligned with its direction of flow. In the case of currents flowing along a rigid floor into a less dense fluid, the case of primary interest here, two mechanisms can produce this instability. The first is gravitational and arises because the nose of the gravity current is elevated above the floor and overrides a buoyantly unstable layer of ambient liquid. The second is a form of viscous fingering analogous to a Saffman–Taylor instability in a Hele-Shaw cell. Whereas the ambient fluid must be more viscous than the current in order for the latter instability to occur, the gravitational instability can occur even if the ambient fluid is less viscous, as long as it is viscous enough to elevate the nose of the current and trap a layer of ambient fluid. For the gravitational mechanism, which is most important when the current and ambient fluids have comparable viscosities, the wavelength when the instability first appears is proportional to a length scale constructed with the viscosity, the flux and the buoyancy. The Saffman–Taylor-type mechanism is most important when the ambient liquid is much more viscous than the current. We have carried out experiments with miscible fluids in a Hele-Shaw cell that show that, at the onset of instability, the ratio of the finger wavelength to the cell width is a constant approximately equal to 2. This result is explained by using the principle that the flow tends to minimize the dissipation associated with the finger perturbation. For the gravity currents with high viscosity ratios, the ratio of the wavelength to the current thickness is also a constant of about 2, apparently consistent with the same mechanism. But, further analysis of this instability mechanism is required in order to assess its role in wavelength selection for gravity currents.

Flow Entrainment of Gravity Currents

2011 ◽  
Author(s):  
K. M. Mok ◽  
Harry H. Yeh ◽  
K. K. Ieong ◽  
K. I. Hoi
Keyword(s):  
Gravity Current ◽  
Three Dimensional ◽  
Propagation Speed ◽  
Bottom Layer ◽  
Gravity Currents ◽  
Flow Structures ◽  
Visualization Technique ◽  
Ambient Fluid ◽  
Actual Flow ◽  
Eddy Generation

The entrainment of gravity currents advancing over a horizontal bed was studied. A two-dimensional rigid-lid flow model was derived assuming ambient-fluid entrainment to the mixing region being supplied only from the bottom layer of the approaching flow. Two sets of laboratory experiments were carried out using the laser-induced fluorescence (LIF) flow visualization technique. With given parameters such as the total fluid depth, densities of the fluids, height of the gravity current head and its propagation speed, and the denser-fluid flow depth behind the head under the mixing region, our model predicts that the thickness of the front flow layer to be entrained is about 35 percents of the height of the gravity current head. Qualitative examination of the flow structures along various planes in the developed fronts suggests that the actual flow structures at the foremost part of the current head are complex and three dimensional. Entrainment of ambient fluid to the current is through various directions starting at its front, which creates an unstable stratification condition there favorable for the subsequent complex three-dimensional eddy generation and growth leading to the formation of the short-crested billows exhibiting the lobe-and-cleft features in the following flow.

Gravity induced vertical motion of dense fluids into saturated granular beds

2021 ◽  
Author(s):  
Rui M L Ferreira ◽  
Gabriel Solis ◽  
Claudia Adduce ◽  
Ana Margarida Ricardo
Keyword(s):  
Porous Layer ◽  
High Speed ◽  
Rectangular Cross Section ◽  
Gravity Current ◽  
Gravity Currents ◽  
Frame Rate ◽  
Ambient Fluid ◽  
Porous Bed ◽  
Loose Packing ◽  
Dense Fluid

<p>Gravity currents propagating over and within porous layers occurs in natural environments and in industrial processes. The particular modes by which the dense fluid flows into the porous layer is a subject that is not sufficiently understood. To overcome this research gap, we conducted laboratory experiments aimed at describing experimentally the dynamics of the drainage flow.</p><p>The experiments were conducted in a horizontal channel with a rectangular cross-section. The channel is 3.0 m long, 0.05 m wide. The porous bottom was composed of 5 cm and 10 cm layers of 3 mm borosilicate spheres – unimodal bed – and of a mixture of 3 mm (50% in weight) and 5 mm spheres (50%) – bi-modal bed. The porosity of the unimodal bed ranged between 0.60 and 0.64 (compatible with loose packing). The porosity of the bi-modal bed ranged between 0.61 and 0.65. All gravity currents were generated by releasing suddenly denser fluid locked by a thin vertical barrier placed at 0.2 m from the channel end. The dense fluid consists in a mixture of freshwater and salt (coloured with Rhodamine) while the ambient fluid is a solution of freshwater and ethanol. The density difference between the ambient fluid and the current, and the need to maintain the same refractive index, determine the amount of salt and alcohol added in each mixture. Here we report the findings of currents with a reduced gravity of 0.06 ms<sup>-2</sup>.</p><p>Each experiment was recorded by an high-speed camera with a frame-rate of 386 Hz and a resolution of 2320 x 1726 pxxpx. Measurements were based on light absorption techniques: a LED light panel 0.3 m high and 0.61 m long was used as back illumination. All images were calibrated to ascribe, pixel by pixel, a concentration value from a 8 bit gray level. Different calibrations were performed for the porous layer and for the surface current.</p><p>Results show that, in the slumping phase, the gravity current flows with velocities compatible with those over rough beds. As the current progresses further attenuation of momentum is noticed owing to mass loss to the porous bed.</p><p>The flow in the porous bed reveals plume instability akin to a Saffman-Taylor instability. The growth of the plumes seems independent from the initial fluid height in both types of porous beds. The wavelength and the growth rate of the plumes depends on the bed material. Plumes grow faster in the case of the bi-modal bed and the wavelength of the bi-modal bed is about 1.5 as that of the unimodal bed. It is hypothesised that the gravity-induced porous flow is best parameterized by a Péclet number defined as a ratio of dispersive (mechanical diffusion) and advective modes of transport. Smaller wavelengths and slower growths are attained for stronger dispersion, characterisitic of the unimodal bed. For bimodal beds, permeability is larger, and thus also advection. This causes the flow to concentrate in faster growing but farther apart plumes.</p><p> </p><p>This research was funded by national funds through Portuguese Foundation for Science and Technology (FCT) project PTDC/CTA-OHR/30561/2017 (WinTherface).</p>

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