Onset Time Of Convection Research Articles

The onset of natural convection in lakes and reservoirs due to night time cooling

A theoretical model, based on linear stability analysis, is proposed to predict the onset of natural convection in lakes and reservoirs due to night time cooling. To such purpose, the system was modelled as a initially quiescent deep Boussinesq fluid reservoir, whose upper boundary temperature changes sinusoidally. From scaling analysis, it is found that critical onset times for convection are proportional to R−2/7, where R is a Rayleigh number including fluid properties and forcing frequency. The proportionality constant was found, from the solution of an eigenvalue problem, as a function of the Prandtl number. The onset time for convection was easily observed from experiments and quantitatively detected as a rapid increase of the RMS of the computed velocity field obtained using PIV. In this controlled conditions, differences close to 10% between predicted and observed times for the start of the convective flow was found. It is apparent from the present set of results that predictions are reasonable.

Onset of convection in a porous medium in the presence of chemical reaction

Using scaling, we show that the stability of a buoyant boundary layer in a porous medium in the presence of a first-order chemical reaction is fully determined by the nondimensional number Da/Ra(2)=k(r)aDϕμ(2)/(kΔρ(0)g)(2), where Da=k(r)aL(Z)(2)/k(r)aL(Z)(2)(Dϕ) is the Damköhler number and Ra=kΔρ(0)gL(Z)/kΔρ(0)gL(Z)(μDϕ) is the solutal Rayleigh number. The time for onset of convection is shown to increase with rising Da/Ra(2). Above a critical Da/DaRa(2)≈2×10(-3) Ra(2)≈2×10(-3), no convection occurs as reaction stabilizes the diffusive layer at a finite thickness. This thickness decreases with increasing Da/Ra(2), becoming zero at Da/Ra(2)≈O(1). As applied to CO(2) geostorage, our results suggest distinct regimes for CO(2) transport in saline aquifers.

High-resolution simulation and characterization of density-driven flow in CO 2 storage in saline aquifers

Simulations are routinely used to study the process of carbon dioxide (CO 2) sequestration in saline aquifers. In this paper, we describe the modeling and simulation of the dissolution–diffusion–convection process based on a total velocity splitting formulation for a variable-density incompressible single-phase model. A second-order accurate sequential algorithm, implemented within a block-structured adaptive mesh refinement (AMR) framework, is used to perform high-resolution studies of the process. We study both the short-term and long-term behaviors of the process. It is found that the onset time of convection follows closely the prediction of linear stability analysis. In addition, the CO 2 flux at the top boundary, which gives the rate at which CO 2 gas dissolves into a negatively buoyant aqueous phase, will reach a stabilized state at the space and time scales we are interested in. This flux is found to be proportional to permeability, and independent of porosity and effective diffusivity, indicative of a convection-dominated flow. A 3D simulation further shows that the added degrees of freedom shorten the onset time and increase the magnitude of the stabilized CO 2 flux by about 25%. Finally, our results are found to be comparable to results obtained from TOUGH2-MP.

Effect of Vertical Heterogeneity on Long-Term Migration of CO2 in Saline Formations

For deep injection of CO2 in thick saline formations, the movements of both the free gas phase and dissolved CO2 are sensitive to variations in vertical permeability. A simple model for vertical heterogeneity was studied, consisting of a random distribution of horizontal impermeable barriers with a given overall volume fraction and distribution of lengths. Analytical results were obtained for the distribution of values for the permeability, and compared to numerical simulations of deep CO2 injection and convection in heterogeneous formations, using multiple realizations for the permeability distribution. It is shown that for a formation of thickness H, the breakthrough times in two dimensions for deep injection scale as H 2 for moderate injection rates. In comparison to heterogeneous shale distributions, a homogeneous medium with equivalent effective vertical permeability has a longer breakthrough time for deep injection, and a longer onset time for convection.

Laboratory Flow Experiments for Visualizing Carbon Dioxide-Induced, Density-Driven Brine Convection

Injection of carbon dioxide (CO2) into saline aquifers confined by low- permeability cap rock will result in a layer of CO2 overlying the brine. Dissolution of CO2 into the brine increases the brine density, resulting in an unstable situation in which more-dense brine overlies less-dense brine. This gravitational instability could give rise to density-driven convection of the fluid, which is a favorable process of practical interest for CO2 storage security because it accelerates the transfer of buoyant CO2 into the aqueous phase, where it is no longer subject to an upward buoyant drive. Laboratory flow visualization tests in transparent Hele-Shaw cells have been performed to elucidate the processes and rates of this CO2 solute-driven convection (CSC). Upon introduction of CO2 into the system, a layer of CO2-laden brine forms at the CO2-water interface. Subsequently, small convective fingers form, which coalesce, broaden, and penetrate into the test cell. Images and time-series data of finger lengths and wavelengths are presented. Observed CO2 uptake of the convection system indicates that the CO2 dissolution rate is approximately constant for each test and is far greater than expected for a diffusion-only scenario. Numerical simulations of our system show good agreement with the experiments for onset time of convection and advancement of convective fingers. There are differences as well, the most prominent being the absence of cell-scale convection in the numerical simulations. This cell-scale convection observed in the experiments may be an artifact of a small temperature gradient induced by the cell illumination.

Effect of Vertical Heterogeneity on Long-Term Migration of CO2 in Saline Formations

Abstract For deep injection of CO2 in thick saline formations, the movements of both the free gas phase and of dissolved CO2 are sensitive to variations in vertical permeability. A simple model for vertical heterogeneity was studied, consisting of a random distribution of horizontal impermeable barriers with a given overall volume fraction and a given distribution of lengths. Analytical results were obtained for the distribution of values for the permeability, and compared with numerical simulations of deep CO2 injection and of convection in heterogeneous formations, using multiple realizations for the permeability distribution. It is shown that for a formation of thickness H, the breakthrough times for deep injection scale as H2 in two dimensions for moderate injection rates. In comparison to heterogeneous shale distributions, a homogenous medium with equivalent effective vertical permeability has a longer breakthrough time for deep injection, and a longer onset time for convection.

Water‐induced convection in the Earth's mantle transition zone

Water enters the Earth's mantle by subduction of oceanic lithosphere. Most of this water immediately returns to the atmosphere through arc volcanism, but a part of it is expected as deep as the mantle transition zone (410–660 km depth). There, slabs can be deflected and linger before sinking into the lower mantle. Because it lowers the density and viscosity of the transition zone minerals (i.e., wadsleyite and ringwoodite), water is likely to affect the dynamics of the transition zone mantle overlying stagnant slabs. The consequences of water exchange between a floating slab and the transition zone are investigated. In particular, we focus on the possible onset of small‐scale convection despite the adverse thermal gradient (i.e., mantle is cooled from below by the slab). The competition between thermal and hydrous effects on the density and thus on the convective stability of the top layer of the slab is examined numerically, including water‐dependent density and viscosity and temperature‐dependent water solubility. For plausible initial water content in a slab (≥0.5 wt %), an episode of convection is likely to occur after a relatively short time delay (5–20 Ma) after the slab enters the transition zone. However, water induced rheological weakening is seen to be a controlling parameter for the onset time of convection. Moreover, small‐scale convection above a stagnant slab greatly enhances the rate of slab dehydration. Small‐scale convection also facilitates heating of the slab, which in itself may prolong the residence time of the slab in the transition zone.

The effect of natural flow of aquifers and associated dispersion on the onset of buoyancy‐driven convection in a saturated porous medium

AbstractCarbon dioxide injection into deep saline aquifers is an important option for managing CO2 emissions. Injected CO2 dissolves into formation brines from above, increasing brine density and creating an unstable hydrodynamic state favorable for natural convection. Long‐term buoyancy‐driven flow of high‐density CO2‐saturated brine leads to faster trapping through improved dissolution and can reduce the risk of CO2 leakage from storage sites. We investigate the role of natural flow of aquifers and associated dispersion on the onset of convection. A linear stability analysis of a transient concentration field in a laterally infinite, horizontal, and saturated porous layer with steady horizontal flow is presented. The layer is subjected to a sudden rise in CO2 concentration from the top and is closed from the bottom. Solution of the stability equations is obtained using a Galerkin technique and the resulting equations are integrated numerically. We found simple scaling relationships that follow tDc∼60(1 + PeT)Ra‐2 for the onset time of convection and a∼0.05Ra/(1 + PeT) for the wavenumber of the initial instabilities. Results reveal that transverse dispersion increases the time to onset of convection for the entire range of Ra. Furthermore, transverse dispersion decreases the critical wavenumber of the instabilities. These results facilitate screening candidate sites for geological CO2 storage. © 2008 American Institute of Chemical Engineers AIChE J, 2009

The onset of convection in fluids with strongly temperature-dependent viscosity cooled from above with implications for planetary lithospheres

The convective instability of a viscous fluid with strongly temperature-dependent viscosity cooled from above provides a basis for better understanding the thermal evolution of the Earth and other planets. The role of temperature-dependent viscosity in the initiation of thermal convection is important since thermally activated creep is strongly temperature dependent. This study introduces a criterion for convective instability based on the ratio of the Rayleigh–Taylor (R-T) growth rate of the thermal density stratification due to conductive cooling and the rate of conductive smoothing of convectively generated temperature variations. This definition of a critical boundary layer Rayleigh number that must be reached for the onset of convection is consistent with numerical experiments reported here as well as earlier theoretical and experimental studies. The length and temperature scales appearing in this Rayleigh number provide valuable physical insight into the behavior of the fluid at the onset of convection. A consistently defined viscosity ratio across the convecting thermal boundary layer leads to a characteristic temperature difference (Δ T c) across this layer. The viscosity ratio inferred from the onset times in numerical experiments reported here (≈10) is similar to that seen in laboratory experiments [Geophys. Res. 99 (1994) 19853] and is consistent with the linearized R-T analysis. Calculation of the onset time of convection and the conductive lid thickness at onset, based on laboratory-derived rheological parameters, provides a better basis to assess the convective instability of the oceanic upper mantle and the possible formation of a thick cratonic lithosphere by the cooling of continental mantle.

3D thermal convection with variable viscosity: can transient cooling be described by a quasi-static scaling law?

The present numerical study describes the transient cooling process of a variable viscosity fluid in the conductive lid regime. The quasi-static hypothesis, assuming that each time-step can be considered as a steady-state heat transfer, is further investigated. A first period of the cooling is essentially transient. Two stages are observed: (1) onset time for convection and (2) convective adjustment period. Onset of convection is seriously delayed when compared to isoviscous thermal convection. The numerical experiments are well described by a boundary layer analysis that we propose, predicting that the dimensionless convective onset time τ o can be scaled as a function of a viscous temperature scale Δ T v linked to the rheological characteristics of the thermal boundary layer [Davaille, A., Jaupart, C., 1993. Transient high-Rayleigh-number thermal convection with large viscosity variations. J. Fluid Mech., 253, 141–166.], the internal Rayleigh number Ra i and the temperature drop Δ T across the fluid layer: τ o ∝Ra i −2/3 ΔT ΔT v 8/3 Due to the slow diffusive heat transfer of the first thermal instabilities through the conductive lid, a convective adjustment stage follows the onset of convection. It is shown that this duration Δ τ II is linked to the square of the conductive lid thickness δ: Δτ II ∝δ 2 Application to the thermal evolution of planetary interiors indicates that this initial transient period is shorter than a few hundreds of million years. The subsequent cooling in the conductive lid regime is shown to be quasi-static. The conductive lid, however, is not equivalent to the elastic lithosphere, as thermal history models usually assume. Cooling numerical experiments described in the present paper are in good agreement with the scaling law describing steady-state heat transfer below a conductive lid.

NUMERICAL STUDY OF CONVECTION HEAT TRANSFER DURING THE MELTING OF ICE IN A POROUS LAYER

A numerical study is made of the melting of ice in a rectangular porous cavity heated from above. The Landau transformation is used to immobilize the ice-water interface, and the Darcy-Boussinesq equations are solved by a finite-difference technique. Results are analyzed in terms of the heating temperature and the aspect ratio of the cavity. A comparison is made with the case of melting from below. It was found that melting from above is more effective than melting from below when the heating temperature is between 0 and 8°C: convection arises earlier, the melting process is faster, and the total melt at steady state is thicker. The critical time for onset of convection is minimum when the upper boundary is heated at 6°C. At this heating temperature, one also obtains a maximum heat transfer rate (Nusselt number).

Transient natural convection heat transfer from a horizontal circular cylinder

The transient natural convection from a horizontal circular cylinder subject to a sudden temperature change was investigated numerically. To solve the stream-function vorticity form of the Navier-Stokes equation and energy equation, the Euler explicit finite difference scheme was used for the transport equation and a noniterative direct elliptic solver for the Poisson equation. An overshoot in the heat transfer coefficient was calculated associated with the conduction-convection transition whose magnitude was influenced by the Pr and Ra numbers. The onset time of convection was nearly proportional to Ra −0.5 in the range 10 3< Ra<10 6 for Pr=0.7 or 7.0. In order to validate the numerical result, an experiment was also conducted from which Schlieren pictures were obtained. They showed good qualitative agreement for the different stages of the plume development.

The onset of natural convection in a fluid layer suddenly heated from below

The onset of convection in a fluid layer in which the temperature profile is developing in time is considered with inclusion of random fluctuations when a constant heat flux is suddenly applied to the bottom surface. Using the expansion in terms of the eigenfunctions of the linear instability, the governing equations are reduced to a set of ordinary random differential equations, which are solved by a Monte Carlo simulation. The dependency of the onset time of convection on the Rayleigh number and the Prandtl number is investigated. Numerical results are in good agreement with available experimental data.

Effect Time-Dependent Heating on the Onset of Rayleigh-Bénard Convection

We consider the problem of the determination of the onset time of Rayleigh-Benard convection in a fluid layer confined between two horizontal planes due to time-dependent heating from below and random fluctuations. Both boundaries are taken to be rigid and perfectly conducting. Using the expansion in terms of eigenfunctions of the stationary Rayleigh-Benard convection, the problem is reduced to a random initial value problem, which is solved by a Monte Carlo technique. Onset times of convection are predicted for various Rayleigh numbers and heating manners. Numerical results are in good agreement with available experimental data.

Onset of thermal convection in a saturated porous medium: experiment and analysis

The onset time of convection in a saturated porous medium heated from below is determined analytically and experimentally. By assuming local thermal equilibrium between the fluid and solid phases, linear amplification theory is applied to the governing equations based on local volume averaging. The parameters of the problem are the Rayleigh number, the Prandtl number and the porous medium shape parameter. The dependency of the onset time on these parameters is examined. The onset time is measured for a system composed of glass beads and water. Four different bead diameters are used. In all cases, relatively good agreement is found between the predicted and measured results. The effect of the depth of the medium on the onset time and the validity of the permeability relation used are discussed.

Transient boiling heat transfer from two different heat sources: Small diameter wire and thin film flat surface on a quartz substrate

Transient boiling heat transfer data are reported for two different heater surface geometries submerged in liquid nitrogen. During the early part of the transient heat pulse, the heat transfer coefficient for both geometries generally agrees with values predicted from classical transient pure conduction equations. The agreement persists until the time for onset of convection which varies approximately as 1 q 2 where q is the heat flux to the fluid.

Numerical experiments on the onset of convective instability in the Earth's mantle

Summary. A simplified model of convection in the mantle is used to investigate the transient effect of cooling a fluid layer from above, The model, representing the mantle overlain by the lithosphere, consists of a two- dimensional fluid layer overlain by a solid conducting lid. The initial temperature of both layers is the same, with the top surface of the lid kept at 0°C throughout. We observe the onset of small-scale flow in the model. In the absence of internal heating the behaviour of the system is controlled by the Rayleigh number, R, and the ratio of the thicknesses of the two layers, a. The onset time of convection as defined by reference to conduction temperature profiles is related simply to a boundary layer critical Rayleigh number. The mean temperature profiles for the convection model are also compared with the observed depth-age relation for oceanic lithosphere and the results are used to estimate the viscosity of the mantle.

Randomly forced Rayleigh-Bénard convection

We consider the onset of Rayleigh–Bénard convection from random fluctuations arising within a fluid. In the specific case in which the fluctuations are thermodynamically determined, we reduce the problem to a random initial value problem for the Fourier modes. For the case of weak nonlinear convection, it is possible to truncate the number of modes and this truncated set is solved both by a Monte Carlo technique and by moment methods for various Rayleigh numbers. We find three stages in the evolution of ordered convection from random fluctuations which correspond to time intervals in which the fluctuations and the nonlinearity have different degrees of importance. It is shown that no simple moment truncation method will succeed and that the time for onset of convection is a mean over a distribution of times for which members of an ensemble exhibit appreciable convective transport.