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.

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>

A Numerical Study of Dean Instability in Non-Newtonian Fluids

2005 ◽  
Vol 128 (1) ◽  
pp. 34-41 ◽  
Author(s):  
H. Fellouah ◽  
C. Castelain ◽  
A. Ould El Moctar ◽  
H. Peerhossaini
Keyword(s):  
Aspect Ratio ◽  
Cross Section ◽  
Power Law ◽  
Numerical Study ◽  
Rectangular Cross Section ◽  
Newtonian Fluids ◽  
Dean Number ◽  
Bingham Number ◽  
Curved Channels ◽  
Power Law Index

We present a numerical study of Dean instability for non-Newtonian fluids in a laminar 180deg curved-channel flow of rectangular cross section. A methodology based on the Papanastasiou model (Papanastasiou, T. C., 1987, J. Rheol., 31(5), pp. 385–404) was developed to take into account the Bingham-type rheological behavior. After validation of the numerical methodology, simulations were carried out (using FLUENT CFD code) for Newtonian and non-Newtonian fluids in curved channels of square or rectangular cross section and for a large aspect and curvature ratios. A criterion based on the axial velocity gradient was defined to detect the instability threshold. This criterion was used to optimize the grid geometry. The effects of curvature and aspect ratio on the Dean instability are studied for all fluids, Newtonian and non-Newtonian. In particular, we show that the critical value of the Dean number decreases with increasing curvature ratio. The variation of the critical Dean number with aspect ratio is less regular. The results are compared to those for Newtonian fluids to emphasize the effect of the power-law index and the Bingham number. The onset of Dean instability is delayed with increasing power-law index. The same delay is observed in Bingham fluids when the Bingham number is increased.

Converging gravity currents over a permeable substrate

2015 ◽  
Vol 778 ◽  
pp. 669-690 ◽  
Author(s):  
Zhong Zheng ◽  
Sangwoo Shin ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Gravity Current ◽  
Gravity Currents ◽  
Numerical Predictions ◽  
Front Position ◽  
Dependent Position ◽  
Self Similar ◽  
Vertical Leakage ◽  
Similar Solutions ◽  
Lock Gate

We study the propagation of viscous gravity currents along a thin permeable substrate where slow vertical drainage is allowed from the boundary. In particular, we report the effect of this vertical fluid drainage on the second-kind self-similar solutions for the shape of the fluid–fluid interface in three contexts: (i) viscous axisymmetric gravity currents converging towards the centre of a cylindrical container; (ii) viscous gravity currents moving towards the origin in a horizontal Hele-Shaw channel with a power-law varying gap thickness in the horizontal direction; and (iii) viscous gravity currents propagating towards the origin of a porous medium with horizontal permeability and porosity gradients in power-law forms. For each of these cases with vertical leakage, we identify a regime diagram that characterizes whether the front reaches the origin or not; in particular, when the front does not reach the origin, we calculate the final location of the front. We have also conducted laboratory experiments with a cylindrical lock gate to generate a converging viscous gravity current where vertical fluid drainage is allowed from various perforated horizontal substrates. The time-dependent position of the propagating front is captured from the experiments, and the front position is found to agree well with the theoretical and numerical predictions when surface tension effects can be neglected.

On Non-Boussinesq Gravity Currents in Non-Rectangular Cross-Section Channels

2014 ◽  
Author(s):  
T. Zemach
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Rectangular Cross Section ◽  
Gravity Current ◽  
Classical Formulation ◽  
Wide Range ◽  
Front Condition ◽  
Significant Generalization ◽  
Self Similar ◽  
Cross Section Geometry

We consider the propagation of a gravity current of density ρc from a lock length x0 and height h0 into an ambient fluid of density ρa in a horizontal channel of height H along the horizontal coordinate x. The bottom and top of the channel are at z = 0, H, and the cross-section is given by the quite general −f1(z) ≤ y ≤ f2(z) for 0 ≤ z ≤ H. When the Reynolds number is large, the resulting flow is governed by the parameters R = ρc/ρa, H* = H/h0 and f(z) = f1(z) + f2(z). We show that the shallow-water one-layer model, combined with a Benjamin-type front condition, provides a versatile formulation for the thickness h and speed u of the current. The results cover in a continuous manner the range of light ρc/ρa ≪ 1, Boussinesq ρc/ρa ≈ 1 and heavy ρc/ρa ≫ 1 currents in a fairly wide range of depth ratio in various cross-section geometries. We obtain analytical solutions for the initial dam-break stage of propagation with constant speed, which appears for any cross-section geometry, and derive explicitly the trend for small and large values of the governing parameters. For large time, t, a self-similar propagation is feasible for f(z) = bzα cross-sections only, with t(2+2α)/(3+2α). The present approach is a significant generalization of the classical non-Boussinesq gravity current problem. The classical formulation for a rectangular (or laterally unbounded) channel is now just a particular case, f(z) = const., in the wide domain of cross-sections covered by this new 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.

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.

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