Fingering Convection Research Articles

An experimental study of density-driven convection of fluid pairs with viscosity contrast in porous media

CO2 dissolution in saline water during aquifer sequestration causes natural convection because of the density difference between CO2-saturated brine and native brine. In this study, the convective fingers behavior during convection in porous media was visualized by magnetic resonance imaging (MRI). Four kinds of miscible fluid pairs were used as the equivalents of CO2-saturated brine and native brine. Images of finger development were obtained. There are two types of convective fingers that form vertically symmetrical shapes: downward fingers of the dense fluid and upward fingers of the light fluid. The typical stages of natural convection, finger appearance, finger propagation, new finger generation, and finger coalescence were observed in this study. We analyzed the effects of viscosity contrast variation on convective onset. It was found that the viscosity contrast affects not only the convection time but also the finger growth velocity. The dimensionless mass transfer coefficient is characterized by the Sherwood number. The Sherwood number has a power law relationship with the Rayleigh number, with the exponent being larger than 1 for fluid pairs with apparent viscosity contrast. Experimental results showed that a higher viscosity contrast of the fluid pair causes a viscosity increase in the transition zone, resulting in lower mobility of the fluids and retarding convective mixing. For fluid pairs with similar densities, a high viscosity contrast is not conducive to efficient mass transfer in the density-driven convection process.

Hydraulic-Head Formulation for Density-Dependent Flow and Transport.

Density-dependent flow and transport solutions for coastal saltwater intrusion investigations, analyses of fluid injection into deep brines, and studies of convective fingering and instabilities of denser fluids moving through less dense fluids typically formulate the groundwater flow equation in terms of pressure or equivalent freshwater head. A formulation of the flow equation in terms of hydraulic head is presented here as an alternative. The hydraulic-head formulation can facilitate adaptation of existing constant-density groundwater flow codes to include density-driven flow by avoiding the need to convert between freshwater head and hydraulic head within the code and by incorporating density-dependent terms as a compartmentalized "correction" to constant-density calculations already performed by the code. The hydraulic-head formulation also accommodates complexities such as unconfined groundwater flow and Newton-Raphson solution schemes more readily than the freshwater-head formulation. Simulation results are presented for four example problems solved using an implementation of the hydraulic-head formulation in MODFLOW.

The Interaction between Shear and Fingering (Thermohaline) Convection

AbstractFingering convection is a turbulent mixing process that can occur in stellar radiative regions whenever the mean molecular weight increases with radius. In some cases, it can have a significant observable impact on stellar structure and evolution. The efficiency of mixing by fingering convection as a standalone process has been studied by Brown et al., but other processes such as rotation, magnetic fields, and shear can affect it. In this paper, we present a first study of the effect of shear on fingering (thermohaline) convection in astrophysics. Using direct numerical simulations, we find that a moderate amount of shear (that is not intrinsically shear unstable) always decreases the mixing efficiency of fingering convection, as a result of the tilt it imparts to the fingering structures. We propose a simple analytical extension of the Brown et al. model in the presence of shear that satisfactorily explains the numerically derived turbulent compositional mixing coefficient for moderate shearing rates and can trivially be implemented in stellar evolution codes. We also measure from the numerical simulations a turbulent viscosity and find that the latter is strongly tied to the turbulent compositional mixing coefficient. Observational implications and caveats of the model are discussed.

Thermohaline layering on the microscale

A theoretical model is developed which illustrates the dynamics of layering instability, frequently realized in ocean regions with active fingering convection. Thermohaline layering is driven by the interplay between large-scale stratification and primary double-diffusive instabilities operating at the microscale – temporal and spatial scales set by molecular dissipation. This interaction is described by a combination of direct numerical simulations and an asymptotic multiscale model. The multiscale theory is used to formulate explicit and dynamically consistent flux laws, which can be readily implemented in large-scale analytical and numerical models. Most previous theoretical investigations of thermohaline layering were based on the flux-gradient model, which assumes that the vertical transport of density components is uniquely determined by their local background gradients. The key deficiency of this approach is that layering instabilities predicted by the flux-gradient model have unbounded growth rates at high wavenumbers. The resulting ultraviolet catastrophe precludes the analysis of such basic properties of layering instability as its preferred wavelength or the maximal growth rate. The multiscale model, on the other hand, incorporates hyperdiffusion terms that stabilize short layering modes. Overall, the presented theory carries the triple advantage of (i) offering an explicit description of the interaction between microstructure and layering modes, (ii) taking into account the influence of non-uniform stratification on microstructure-driven mixing, and (iii) avoiding unphysical behaviour of the flux-gradient laws at small scales. While the multiscale approach to the parametrization of time-dependent small-scale processes is illustrated here on the example of fingering convection, we expect the proposed technique to be readily adaptable to a wide range of applications.

Enhanced Mixing in Magnetized Fingering Convection, and Implications for Red Giant Branch Stars

Abstract Double-diffusive convection has been well studied in geophysical contexts, but detailed investigations of the regimes characteristic of stellar or planetary interiors have only recently become feasible. As most astrophysical fluids are electrically conducting, it is possible that magnetic fields play a role in either enhancing or suppressing double-diffusive convection, but to date there have been no numerical investigations of such possibilities. Here we study the effects of a vertical background magnetic field (aligned with the gravitational axis) on the linear stability and nonlinear saturation of fingering (thermohaline) convection, through a combination of theoretical work and direct numerical simulations. We find that a vertical magnetic field rigidifies the fingers along the vertical direction, which has the remarkable effect of enhancing vertical mixing. We propose a simple analytical model for mixing by magnetized fingering convection, and argue that magnetic effects may help explain discrepancies between theoretical and observed mixing rates in low-mass red giant branch stars. Other implications of our findings are also discussed.

Differential Diffusivity Effects in Reactive Convective Dissolution

When a solute A dissolves into a host fluid containing a reactant B, an A + B → C reaction can influence the convection developing because of unstable density gradients in the gravity field. When A increases density and all three chemical species A, B and C diffuse at the same rate, the reactive case can lead to two different types of density profiles, i.e., a monotonically decreasing one from the interface to the bulk and a non-monotonic profile with a minimum. We study numerically here the nonlinear reactive convective dissolution dynamics in the more general case where the three solutes can diffuse at different rates. We show that differential diffusion can add new dynamic effects like the simultaneous presence of two different convection zones in the host phase when a non-monotonic profile with both a minimum and a maximum develops. Double diffusive instabilities can moreover affect the morphology of the convective fingers. Analysis of the mixing zone, the reaction rate, the total amount of stored A and the dissolution flux further shows that varying the diffusion coefficients of the various species has a quantitative effect on convection.

Convective dissolution of CO2 in water and salt solutions

Dissolution of CO2 into saline aquifers can lead to the development of buoyancy-driven convection in the brine which enhances the efficiency of CO2 transfer. We analyze here experimentally the onset, development and dynamic properties of such convective fingering of CO2 into water, Antarctic water and in NaCl salt solutions of various concentrations to study the influence of varying the salt concentration on the buoyancy-driven convective dynamics. The convective dissolution pattern is visualized with the help of a schlieren imaging system sensitive to density gradients in the solution. We quantify the growth of convective fingers by performing, among others, a Fourier analysis of the pattern formation at early times and qualitatively study the nonlinear spatio-temporal dynamics at later times. In agreement with theoretical predictions, we find that increasing the salt concentration hinders the development of the instability as it delays the onset of convection, increases the wavelength of the convective pattern, decreases the growth rate and velocity of fingers as well as their interactions. Our experimental results provide quantitative data that should help the benchmarking of theoretical studies.

Qualitative and quantitative experimental study of convective mixing process during storage of CO2 in heterogeneous saline aquifers

CO2 dissolution is considered as one of the most promising mechanisms for trapping of free-phase CO2 into brine. It causes an increased density of the brine and initiation of gravitational instability that eventually leads to density-driven natural convection in saline aquifers. Correct estimation of the onset time for convection and the rate of dissolution of CO2 into brine is important because the timescale for dissolution corresponds to the timescale over which free-phase CO2 has a chance to leak out. The gravitational instability of a diffusive boundary layer in porous media has been studied in several papers in recent years, but there are few works about the behavior of density-driven natural convection mechanism in heterogeneous saline aquifers. Barriers such as shales and calcites layers are common types of heterogeneities in geological formations that are important in the fluid flow. Despite the recognized importance of convective dissolution in these heterogeneous geological formations, there is no experimental data available for studying the accelerated mass transfer rate of CO2 into these media.In this paper, we investigated the effect of the regular distribution of barriers on the rate of dissolution of CO2 into water and geometries of convection fingers. A series of experiments were performed using a precise experimental set-up with barrier heterogeneous Hele-Shaw cell geometries and by using CO2 and water. The approach and procedure for performing the experiments give us this opportunity to have both qualitative (images and movies) and quantitative (amount of the dissolved CO2 into water) data at the same time. The behavior of convection pattern after onset time and the effect of system properties on the behavior of convective mixing process will be presented and discussed. Moreover, some speeded-up movies from the experiments that are suitable for improving public awareness of the problem have been uploaded on the internet platform. Lastly, the relationships between dissolution flux after onset time for convection and barrier properties are discussed.

Qualitative and quantitative experimental study of convective mixing process during storage of CO2 in homogeneous saline aquifers

The storage of CO2 in deep formations like saline aquifers is being actively considered in order to support the reduction of greenhouse gas emissions. It has been observed that dissolution of CO2 into brine causes an increased density in the mixture and if the corresponding Rayleigh number of the porous medium is high enough to initiate convection flows, then density-driven natural convection occurs and the rate of dissolution increases. As such, an increased contribution by the dissolution mechanism for trapping of CO2 decreases the risk of leakage.However, despite the recognized importance of convective dissolution in geological CO2 storage, there is insufficient experimental data available for studying the accelerated mass transfer rate of CO2 into saline aquifers. In this paper, we focus on performing a series of novel experiments about the density-driven natural convection mechanism in a precise experimental set-up with homogeneous Hele-Shaw cell geometries and by using CO2 and water. Our new approach and procedure for performing the experiments give us an opportunity to have both qualitative (images and video) and quantitative (amount of dissolved CO2 into water) data at the same time. This study examines onset time for convection, critical wavelength of convection fingers and dissolution flux of CO2 into water as objective parameters. The growth and progress of convection fingers after onset time for convection and the effect of model properties on the behavior of the convective mixing process are presented and discussed. Moreover, there are some speeded-up videos from the experiments that are suitable for improving public awareness of the problem facing society. Based on the performed experiments, the calculated dimensionless onset times for convection are 194.90 and 227.80 from quantitative and qualitative measurements respectively and the calculated critical wavelength of convection fingers is 2π/0.0524. Additionally, the dissolution flux of CO2 into water after onset time for convection is related linearly to Δρ·g·cos(θ)·k·C0/μ by a prefactor of 0.021. These results are comparable with the available theoretical and numerical works in the literature. Comparison of the results of the experimental models and their equivalent numerical models shows that while the numerical models are good tools for prediction of growth mechanism, shapes and positions of convection fingers, their effectiveness in the prediction of the onset times for convection and the dissolution fluxes is uncertain.

Dissipative structures: From reaction-diffusion to chemo-hydrodynamic patterns.

The interplay of reaction and diffusion processes can trigger localized spatiotemporal patterns when two solutions containing separate reactants A and B of an oscillating reaction are put in contact. Using the Brusselator, a classical model for chemical oscillations, we show numerically that localized waves and Turing patterns as well as reaction-diffusion (RD) patterns due to an interaction between these two kinds of modes can develop in time around the reactive contact zone depending on the initial concentration of reactants and diffusion coefficients of the intermediate species locally produced. We further explore the possible hydrodynamic destabilization of an initially buoyantly stable stratification of such an A + B → oscillator system, when the chemical reaction provides a buoyant periodic forcing via localized density changes. Guided by the properties of the underlying RD dynamics, we predict new chemo-hydrodynamic instabilities on the basis of the dynamic density profiles which are here varying with the concentration of one of the intermediate species of the oscillator. Nonlinear simulations of the related reaction-diffusion-convection equations show how the active coupling between the localized oscillatory kinetics and buoyancy-driven convection can induce pulsatile convective fingering and pulsatile plumes as well as rising or sinking Turing spots, depending on the initial concentration of the reactants and their contribution to the density.

Dispersion-driven instability of mixed convective flow in porous media

This paper investigates the impact of hydrodynamic dispersion on the stability of free convection in a saturated horizontal porous layer subject to a transient vertical concentration gradient and a steady horizontal background flow. A linear stability analysis (LSA) was conducted using the quasi-steady-state approximation to obtain neutral stability curves, critical times, and the corresponding wavenumbers as a function of dispersivity ratio (α) and longitudinal dispersion strength (β). The LSA results showed that the dispersive boundary layer becomes less unstable as longitudinal and transverse dispersivity increase. In addition, for the isotropic dispersive system with α = 1, the critical time and its corresponding wavenumber follow τc = 167.6/(1 − β) and κc = 0.0696 (1 − β), respectively. The nonlinear dynamics of the system were studied by examining the interaction of convective fingers, dissolution flux, and the time-dependent Sherwood number. Finally, the results were applied to 24 deep saline aquifers in the Alberta Basin.

Experimental Study of Density-driven Convection in Porous Media by Using MRI

Density-driven convection is known as one of the physical phenomena for CO2 injection in saline aquifers, which accelerates the transfer of CO2 into the groundwater, increases the dissolution rate and shorten the dissolution time, thus it is favourable for long-term CO2 injection and the practical interest of storage security. Understanding how the occurrence and form of convective fingers is crucial to understanding the role of that Density-driven-convection behaviour. The purpose of this paper is to design a visual experimental set-up based on the MRI technique and present new analogue fluid pairs that reconstructed the convective process. The density-driven convection was formed by two miscible liquids at the interface demonstrated by fingers. We observed the development and growth of density-driven fingers of two-phase fluid i.e., water/ deuterium oxide, and brine/ deuterium oxide at different permeability porous media. Experiments indicated that the relationship of convection onset time to Rayleigh number changed with the density difference and permeability. We present an experimental study for the Rayleigh number in the range 583.279 to 2054.111. The results show that the higher permeability and density difference is conducive to shorten the convection onset time. The motivation behind this results is related to the implications for predicts the CO2 dissolution trapping time and accumulations in geological storage of carbon dioxide.

Importance of fingering convection for accreting white dwarfs in the framework of full evolutionary calculations: the case of the hydrogen-rich white dwarfs GD 133 and G 29-38

A large fraction of white dwarf stars shows photospheric chemical composition polluted by heavy elements accreted from a debris disk. Such debris disks result from the tidal disruption of rocky planetesimals which had survived to whole stellar evolution from the main sequence to the final white dwarf stage. Determining the accretion rate of this material is an important step towards estimating the mass of the planetesimals and towards understanding the ultimate fate of the planetary systems. The accretion of heavy material with a mean molecular weight, $\mu$, higher than the mean molecular weight of the white dwarf outer layers, induces a double-diffusive instability producing fingering convection and extra-mixing. As a result, the accreted material is diluted deep into the star. We explore the effect of this extra-mixing on the abundance evolution of Mg, O, Ca, Fe and Si in the cases of the two well studied polluted DAZ white dwarfs: GD~133 and G~29-38. We performed numerical simulations of the accretion of material with a chemical composition similar to the bulk Earth one. We considered accretion rates from $10^{4}$~g/s to $10^{10}$~g/s. The double-diffusive instability develops on a very short time scale. The accretion rate needed to reproduce the observed abundances exceeds by more than 2 orders of magnitude the rate estimated by neglecting the fingering convection in the case of GD~133, and by approximately 1.7 dex in the case of G~29-38. Our numerical simulations show that fingering convection is an efficient mechanism to mix the accreted material and that it must be taken into account in the determination of accretion rates.

Layer formation in sedimentary fingering convection

When particles settle through a stable temperature or salinity gradient they can drive an instability known as sedimentary fingering convection. This phenomenon is thought to occur beneath sediment-rich river plumes in lakes and oceans, in the context of marine snow where decaying organic materials serve as the suspended particles or in the atmosphere in the presence of aerosols or volcanic ash. Laboratory experiments of Houk & Green (Deep-Sea Res., vol. 20, 1973, pp. 757–761) and Green (Sedimentology, vol. 34(2), 1987, pp. 319–331) have shown sedimentary fingering convection to be similar to the more commonly known thermohaline fingering convection in many ways. Here, we study the phenomenon using three-dimensional direct numerical simulations. We find evidence for layer formation in sedimentary fingering convection in regions of parameter space where it does not occur for non-sedimentary systems. This is due to two complementary effects. Sedimentation affects the turbulent fluxes and broadens the region of parameter space unstable to the $\unicode[STIX]{x1D6FE}$-instability (Radko, J. Fluid Mech., vol. 497, 2003, pp. 365–380) to include systems at larger density ratios. It also gives rise to a new layering instability that exists in $\unicode[STIX]{x1D6FE}$-stable regimes. The former is likely quite ubiquitous in geophysical systems for sufficiently large settling velocities, while the latter probably grows too slowly to be relevant, at least in the context of sediments in water.

Floquet Instability of Gravity-Modulated Salt Fingering in a Porous Medium

Fingering convection in a horizontal doubly stratified fluid-saturated porous medium is examined under the influence of modulating gravitational field. The Brinkman model of momentum transfer is considered, and the Boussinesq approximation is assumed for natural convection. Floquet theory is employed to determine the onset condition within the framework of linear theory. Continued fractions that can handle arbitrary modulational parameters has been used to predict the dynamic instability regions at the marginal state and critical instability boundaries. Certain combinations of solutal Rayleigh and Schmidt numbers give rise to the occurrence of doubly unstable regions. An increase in the modulation amplitude always encourages instability. The competition between synchronous and subharmonic instability modes is observed for a certain range of values of the parameters. Solutal convection is found to introduce closed disconnected instability loops at the marginal state. It is seen that gravity modulation effe...

Convective mixing fingers and chemistry interaction in carbon storage

Dissolution of carbon-dioxide into formation fluids during carbon capture and storage (CCS) can generate an instability with a denser CO2-rich fluid located above the less dense native aquifer fluid. This instability promotes convective mixing, enhancing CO2 dissolution and favouring the storage safety. Convective mixing has been extensively analysed in the context of CCS over the last decade, however the interaction between convective mixing and geochemistry has been insufficiently addressed. This relation is explored using a fully coupled model taking into account the porosity and permeability variations due to dissolution-precipitation reactions in a realistic geochemical system based on the Hontomín (Spain) potential CCS site project. This system, located in a calcite, dolomite, and gypsum bearing host rock, has been analysed for a variety of Rayleigh and Damköhler values. Results show that chemical reactions tend to enhance CO2 dissolution. The model illustrates the first stages of porosity channel development, demonstrating the significance of fluid mixing in the development of porosity patterns. The influence of non-carbon species on CO2 dissolution shown in this study demonstrates the needs for realistic chemical and kinetic models to ensure the precision of physical models to accurately represent the carbon-dioxide injection process.

Stability analysis of dissolution-driven convection in porous media

We study the stability of dissolution-driven convection in the presence of a capillary transition zone and hydrodynamic dispersion in a saturated anisotropic porous medium, where the solute concentration is assumed to decay via a first-order chemical reaction. While the reaction enhances stability by consuming the solute, porous media anisotropy, hydrodynamic dispersion, and capillary transition zone destabilize the diffusive boundary layer that is unstably formed in a gravitational field. We perform linear stability analysis, based on the quasi-steady-state approximation, to assess critical times, critical wavenumbers, and neutral stability curves as a function of anisotropy ratio, dispersivity ratio, dispersion strength, material parameter, Bond number, Damköhler number, and Rayleigh number. The results show that the diffusive boundary layer becomes unstable in anisotropic porous media where both the capillary transition zone and dispersion are considered, even if the geochemical reaction is significantly large. Using direct numerical simulations, based on the finite difference method, we study the nonlinear dynamics of the system by examining dissolution flux, interaction of convective fingers, and flow topology. The results of nonlinear simulations confirm the predictions from the linear stability analysis and reveal that the fingering pattern is significantly influenced by combined effects of reaction, anisotropy, dispersion, and capillarity. Finally, we draw conclusions on implications of our results on carbon dioxide sequestration in deep saline aquifers.

Physical Properties of Persian Gulf Outflow Thermohaline Intrusion in the Oman Sea

Various CTD data obtained in the Oman Sea are analyzed to explain structural features of intrusive layering. Special attention is compensated to thermohaline intrusions observed in layers (depth ranges of 150 m to 450 m, 150 m to 350 m, 100 m to 350 m and 150 m and 400 m in the winter, spring, summer and autumn, respectively). The temperature and salinity profiles in thermohaline intrusion have sawtooth structure in all stations, while they have step structure in density field. Based on interpretations, detailed estimates of thickness are about 10 to 20 meters. The T-S diagrams show the positions of the outflow intrusion with different thicknesses and depths for all seasons in the Oman Sea. Vertical profiles of temperature and salinity show two boundaries in the upper and lower layers. They are prone to double diffusive convection. Salt fingering and diffusive convection can be seen in both the upper and lower boundaries, and salt fingering is stronger at the lower boundary. Diffusive convection also is visible from the surface to the mid-depth of the plume outflow, and the diffusive intrusion is more severe at the upper boundary than the surface and deep. The intensity of double diffusion in the bottom border is greater than the upper boundary. At the deeper parts, the stratification is completely stable. Variations of the positions of greatest salinities in different diagrams are due to changing water masses for different locations and depths and paths of intrusive flow.

Double-diffusive lock-exchange gravity currents

Double-diffusive lock-exchange gravity currents in the fingering regime are explored via two- and three-dimensional Navier–Stokes simulations in the Boussinesq limit. Even at modest Reynolds numbers, for which single-diffusive gravity currents remain laminar, double-diffusive currents are seen to give rise to pronounced small-scale fingering convection. The front velocity of these currents exhibits a non-monotonic dependence on the diffusivity ratio and the initial stability ratio. Strongly double-diffusive currents lose both heat and salinity more quickly than weakly double-diffusive ones, and they lose salinity more quickly than heat, so that the density difference driving them increases. This differential loss of heat and salinity furthermore results in the emergence of strong local density maxima and minima along the top and bottom walls in the gate region, which in turn promote the formation of secondary, counterflowing currents along the walls. These secondary currents cause the flow to develop a three-layer structure. The late stages of the flow are dominated by currents flowing oppositely to the original ones. Three-dimensional simulation results are consistent with the trends observed in a two-dimensional parametric study. A detailed analysis of the energy budget demonstrates that strongly double-diffusive currents can release several times their initially available potential energy, and convert large amounts of internal energy into mechanical energy via scalar diffusion. Scaling arguments based on the simulation results suggest that even low Reynolds number double-diffusive gravity currents are governed by a balance of buoyancy and turbulent drag.

Schlieren imaging of viscous fingering in a horizontal Hele-Shaw cell

Interfaces between different fluids can be unstable with regard to hydrodynamic instabilities such as viscous fingering or buoyancy-driven convection. To study such instabilities experimentally for transparent fluids, dyes or chemical indicators are most often used to track the dynamics. While the interfacial deformation can easily be tracked by color changes, it is difficult to have access to the internal flow structure for comparison with theoretical predictions. To overcome this problem, a modification of a Schlieren technique is introduced to image 3D flows during viscously driven instabilities in a horizontal Hele-Shaw cell without using any dye or chemical indicator. The method is exquisitely sensitive, readily yielding information about 3D flows in gaps under a millimeter and allowing imaging of the flow structure internal to the fingers, rather than merely imaging the flow boundary. Following a description of the technique, visualization of dynamics for nonreactive water–glycerol and reactive displacements is presented revealing previously unobserved internal flows. These flows are tentatively interpreted in terms of known theoretical predictions.