Sediment-laden fresh water above salt water: linear stability analysis

2011 ◽  
Vol 691 ◽  
pp. 279-314 ◽  
Author(s):  
P. Burns ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Fresh Water ◽  
Settling Velocity ◽  
Interface Thickness ◽  
Particle Settling ◽  
Salinity Field ◽  
Unstable Layer ◽  
Stability Results ◽  
Double Diffusive

AbstractWhen a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

2002 ◽  
Vol 451 ◽  
pp. 261-282 ◽  
Author(s):  
F. GRAF ◽  
E. MEIBURG ◽  
C. HÄRTEL
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Linear Stability ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interface Thickness ◽  
Gap Width ◽  
Stability Results ◽  
Hele Shaw Cell

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

Sediment-laden fresh water above salt water: nonlinear simulations

2014 ◽  
Vol 762 ◽  
pp. 156-195 ◽  
Author(s):  
P. Burns ◽  
E. Meiburg
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Fresh Water ◽  
Linear Stability Analysis ◽  
Settling Velocity ◽  
Taylor Instability ◽  
Interfacial Region ◽  
Rayleigh Taylor Instability ◽  
Sediment Interface ◽  
Double Diffusive

AbstractWhen a layer of particle-laden fresh water is placed above clear, saline water, both double-diffusive and Rayleigh–Taylor instabilities may arise. The present investigation extends the linear stability analysis of Burns & Meiburg (J. Fluid Mech., vol. 691, 2012, pp. 279–314) into the nonlinear regime, by means of two- and three-dimensional direct numerical simulations (DNS). The initial instability growth in the DNS is seen to be consistent with the dominant modes predicted by the linear stability analysis. The subsequent vigorous growth of individual fingers gives rise to a secondary instability, and eventually to the formation of intense plumes that become detached from the interfacial region. The simulations show that the presence of particles with a Stokes settling velocity modifies the traditional double-diffusive fingering by creating an unstable ‘nose region’ in the horizontally averaged profiles, located between the upward-moving salinity and the downward-moving sediment interface. The effective thickness $l_{s}$ ($l_{c}$) of the salinity (sediment) interface grows diffusively, as does the height $H$ of the nose region. The ratio $H/l_{s}$ initially grows and then plateaus, at a value that is determined by the balance between the flux of sediment into the rose region from above, the double-diffusive/Rayleigh–Taylor flux out of the nose region below, and the rate of sediment accumulation within the nose region. For small values of $H/l_{s}\leqslant O(0.1)$, double-diffusive fingering dominates, while for larger values $H/l_{s}\geqslant O(0.1)$ the sediment and salinity interfaces become increasingly separated in space and the dominant instability mode becomes Rayleigh–Taylor like. A scaling analysis based on the results of a parametric study indicates that $H/l_{s}$ is a linear function of a single dimensionless grouping that can be interpreted as the ratio of inflow and outflow of sediment into the nose region. The simulation results furthermore indicate that double-diffusive and Rayleigh–Taylor instability mechanisms cause the effective settling velocity of the sediment to scale with the overall buoyancy velocity of the system, which can be orders of magnitude larger than the Stokes settling velocity. While the power spectra of double-diffusive and Rayleigh–Taylor-dominated flows are qualitatively similar, the difference between flows dominated by fingering and leaking is clearly seen when analysing the spectral phase shift. For leaking-dominated flows a phase-locking mechanism is observed, which intensifies with time. Hence, the leaking mode can be interpreted as a fingering mode which has become phase-locked due to large-scale overturning events in the nose region, as a result of a Rayleigh–Taylor instability.

scholarly journals A settling-driven instability in two-component, stably stratified fluids

2017 ◽  
Vol 816 ◽  
pp. 243-267 ◽  
Author(s):  
A. Alsinan ◽  
E. Meiburg ◽  
P. Garaud
Keyword(s):  
Linear Stability ◽  
Settling Velocity ◽  
Scalar Fields ◽  
State Density ◽  
Stratified Fluids ◽  
Instability Mechanism ◽  
Two Component ◽  
Constant Gradient ◽  
Stability Results ◽  
Double Diffusive

We analyse the linear stability of stably stratified fluids whose density depends on two scalar fields where one of the scalar fields is unstably stratified and involves a settling velocity. Such conditions may be found, for example, in flows involving the transport of sediment in addition to heat or salt. A linear stability analysis for constant-gradient base states demonstrates that the settling velocity generates a phase shift between the perturbation fields of the two scalars, which gives rise to a novel, settling-driven instability mode. This instability mechanism favours the growth of waves that are inclined with respect to the horizontal. It is active for all density and diffusivity ratios, including for cases in which the two scalars diffuse at identical rates. If the scalars have unequal diffusivities, it competes with the elevator mode waves of the classical double-diffusive instability. We present detailed linear stability results as a function of the governing dimensionless parameters, including for lateral gradients of the base state density fields that result in predominantly horizontal intrusion instabilities. Highly resolved direct numerical simulation results serve to illustrate the nonlinear competition of the various instabilities for such flows in different parameter regimes.

scholarly journals ANÁLISIS HIDROGRÁFICO E ICTIOLÓGICO DE LAS CAPTURAS REALIZADAS CON UNA RED DE TRAMPA FIJA EN LA LAGUNA DE GANDOCA, LIMÓN, COSTA RICA

2010 ◽  
Vol 2 ◽  
pp. 9
Author(s):  
Rosario Benavides Morera ◽  
Carlos L. Brenes
Keyword(s):  
Costa Rica ◽  
Fresh Water ◽  
Annual Cycle ◽  
Tidal Wave ◽  
Maximum Temperature ◽  
Water Runoff ◽  
Direct Influence ◽  
Salinity Field ◽  
First Maturity ◽  
The Relationship

Se realizó un estudio ictiológico de las capturas obtenidas con una red de trampa fija y un registro de las  propiedades hidrológicas en la Laguna de Gandoca, Limón, entre abril del 2006 y julio del 2007. Se identificaron 13 especies pertenecientes a 10 familias. El 98% de las capturas estuvo conformado por cuatro especies de valor comercial: Centropomus pectinatus (77%), Eucinostomus gula (9%), Caranx latus (7%) y Stellifer colonensis (5%). C. pectinatus (róbalo) estuvo presente en todas las capturas. Para C. pectinatus, se determinó que la relación longitud total - peso se ajustó potencialmente a través de la ecuación Pt= 0.004Lt3.1848. La talla de primera madurez de los róbalos se determinó en 30 cm de longitud total. En el ciclo anual de las propiedades termohalinas superficiales, la temperatura máxima se registró en el mes de setiembre (32°C) y la mínima durante diciembre (25.5°C). La laguna exhibe sus mayores salinidades en octubre (21), mientras que las salinidades menores ocurrió en julio y diciembre (1). Entre setiembre y noviembre, la influencia de la onda mareal se extendió de hasta media laguna con salinidades de 20, mientras que en la parte más interna no excedió a 5. Los valores más altos de contenido de oxígeno se observaron entre setiembre y noviembre, cuando el aporte de agua dulce proveniente de las escorrentías es mínimo. Finalmente, las características espacio-temporales del campo salino tienen una influencia directa en la composición y distribución de la ictiofauna que habita en la laguna.An ichthyological study of the fishing catch in a fixed trap net along with hydrographic sampling of the hydrological properties in Gandoca Lagoon, Limón, was carried out between April 2006 and July 2007. Thirteen species belonging to 10 families were identified. Ninty-eight percent of the captures belong to four species of commercial value: Centropomus pectinatus (77%), Eucinostomus gula (9%), Caranx latus (7%) and Stellifer colonensis (5%). C. pectinatus was present in all captures. For C. pectinatus (snook), the relationship between total length and weight was adjusted potentially through the equation Pt= 0.004Lt3.1848. The size of first maturity of snooks was 30 cm. The annual cycle of surface termohaline properties shows the maximum temperature in September (32°C) and the minimum during December (25.5°C). The Lagoon exhibits maximum salinities in October (21) and minimum in July and December (1). Between September and November the influence of the tidal wave extends to the middle of the lagoon with salinities of 20, while in the inner part it does not exceed 5. The highest values in oxygen content were observed between September and November, when the contribution of fresh water runoff is minimal. The space-time characteristics of the salinity field have a direct influence on the composition and distribution of the icthyofauna that inhabits the Lagoon.

Comparison of linear stability results with flight transition data

AIAA Journal ◽  
1995 ◽  
Vol 33 (1) ◽  
pp. 161-163 ◽  
Author(s):  
J. A. Masad ◽  
M. R. Malik
Keyword(s):  
Linear Stability ◽  
Stability Results

scholarly journals Effects of Bt maize on the reproduction and development of the freshwater snail Bulinus tropicus

2012 ◽  
Vol 31 (1) ◽  
Author(s):  
K. Minnaar ◽  
H. Bouwman
Keyword(s):  
Growth Rate ◽  
Fresh Water ◽  
Agricultural Land ◽  
Genetically Modified ◽  
Genetically Modified Crops ◽  
Freshwater Snail ◽  
Bt Maize

This study was conducted to investigate the effects of genetically modified crops onaquatic environments around agricultural land. Fresh water snails indicated effects duringembryonic development, and on the growth rate of survivors.

scholarly journals Influence of Bottom Topography on Vortex Stability

2019 ◽  
Vol 49 (12) ◽  
pp. 3199-3219 ◽  
Author(s):  
Bowen Zhao ◽  
Emma Chieusse-Gérard ◽  
Glenn Flierl
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Bottom Topography ◽  
Relative Vorticity ◽  
Subsequent Evolution ◽  
Vortex Stability ◽  
Number Of Layers ◽  
One Step ◽  
Baroclinic Vortices ◽  
Dynamics Simulations

AbstractThe effects of topography on the linear stability of both barotropic vortices and two-layer, baroclinic vortices are examined by considering cylindrical topography and vortices with stepwise relative vorticity profiles in the quasigeostrophic approximation. Four vortex configurations are considered, classified by the number of relative vorticity steps in the horizontal and the number of layers in the vertical: barotropic one-step vortex (Rankine vortex), barotropic two-step vortex, and their two-layer, baroclinic counterparts with the vorticity steps in the upper layer. In the barotropic calculation, the vortex is destabilized by topography having an oppositely signed potential vorticity jump while stabilized by topography of same-signed jump, that is, anticyclones are destabilized by seamounts while stabilized by depressions. Further, topography of appropriate sign and magnitude can excite a mode-1 instability for a two-step vortex, especially relevant for topographic encounters of an otherwise stable vortex. The baroclinic calculation is in general consistent with the barotropic calculation except that the growth rate weakens and, for a two-step vortex, becomes less sensitive to topography (sign and magnitude) as baroclinicity increases. The smaller growth rate for a baroclinic vortex is consistent with previous findings that vortices with sufficient baroclinic structure could cross the topography relatively easily. Nonlinear contour dynamics simulations are conducted to confirm the linear stability analysis and to describe the subsequent evolution.

Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis

2013 ◽  
Vol 721 ◽  
pp. 268-294 ◽  
Author(s):  
L. Talon ◽  
N. Goyal ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Variable Density ◽  
Local Concentration ◽  
Miscible Displacements ◽  
Instability Modes ◽  
Finger Tip

AbstractA computational investigation of variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells is presented. As a first step, two-dimensional base states are obtained by means of simulations of the Stokes equations, which are nonlinear due to the dependence of the viscosity on the local concentration. Here, the vertical position of the displacement front is seen to reach a quasisteady equilibrium value, reflecting a balance between viscous and gravitational forces. These base states allow for two instability modes: first, there is the familiar tip instability driven by the unfavourable viscosity contrast of the displacement, which is modulated by the presence of density variations in the gravitational field; second, a gravitational instability occurs at the unstably stratified horizontal interface along the side of the finger. Both of these instability modes are investigated by means of a linear stability analysis. The gravitational mode along the side of the finger is characterized by a wavelength of about one half to one full gap width. It becomes more unstable as the gravity parameter increases, even though the interface is shifted closer to the wall. The growth rate is largest far behind the finger tip, where the interface is both thicker, and located closer to the wall, than near the finger tip. The competing influences of interface thickness and wall proximity are clarified by means of a parametric stability analysis. The tip instability mode represents a gravity-modulated version of the neutrally buoyant mode. The analysis shows that in the presence of density stratification its growth rate increases, while the dominant wavelength decreases. This overall destabilizing effect of gravity is due to the additional terms appearing in the stability equations, which outweigh the stabilizing effects of gravity onto the base state.

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