20 results
A shallow approximation for ice streams sliding over strong beds
- Katarzyna L. P. Warburton, Duncan R. Hewitt, Colin R. Meyer, Jerome A. Neufeld
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- Journal:
- Journal of Glaciology , First View
- Published online by Cambridge University Press:
- 25 July 2023, pp. 1-12
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Ice streams are regions of rapid ice sheet flow characterised by a high degree of sliding over a deforming bed. The shallow shelf approximation (SSA) provides a convenient way to obtain closed-form approximations of the velocity and flux in a rapidly sliding ice stream when the basal drag is much less than the driving stress. However, the validity of the SSA approximation breaks down when the magnitude of the basal drag increases. Here we find a more accurate expression for the velocity and flux in this transitional regime before vertical deformation fully dominates, in agreement with numerical results. The closed-form expressions we derive can be incorporated into wider modelling efforts to yield a better characterisation of ice stream dynamics, and inform the use of the SSA in large-scale simulations.
Obstructed free-surface viscoplastic flow on an inclined plane
- Edward M. Hinton, Duncan R. Hewitt, Andrew J. Hogg
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- Journal:
- Journal of Fluid Mechanics / Volume 964 / 10 June 2023
- Published online by Cambridge University Press:
- 02 June 2023, A35
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The interaction of steady free-surface flows of viscoplastic material with a surface-piercing obstruction of square cross-section on an inclined plane is investigated theoretically. The flow thickness increases upstream of the obstruction and decreases in its lee. The flow depends on two dimensionless parameters: an aspect ratio that relates the flow thickness, the obstruction width and the plane inclination; and a Bingham number that quantifies the magnitude of the yield stress relative to the gravitationally induced stresses. Flows with a non-vanishing yield stress always form a static ‘dead’ zone in a neighbourhood of the upstream and downstream stagnation points. For relatively wide obstructions, a deep ‘ponded’ region develops upstream with a small dead zone, while the deflected flow reconnects over relatively long distances downstream. The depth of the upstream pond increases with both the dimensionless yield stress and width of the obstruction, while the unyielded dead zone varies primarily with the yield stress. Both are predicted asymptotically by balancing the volume flux of fluid into and out of the ponded region. When the obstruction is narrow, the perturbation to the depth of the oncoming flow is reduced. It exhibits fore–aft antisymmetry, while the dead zone is symmetric to leading order. Increasing the yield stress leads to larger dead zones that eventually encompass all of the upstream- and downstream-facing boundaries of the obstruction and fully divert the flow. Results for obstructions with circular and rhomboidal cross-sections are also presented and illustrate the effects of boundary shape on the properties of the steady flow.
Clogged corners
- Duncan R. Hewitt
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- Journal:
- Journal of Fluid Mechanics / Volume 948 / 10 October 2022
- Published online by Cambridge University Press:
- 02 September 2022, F1
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Viscoplastic materials cannot flow if the local stress falls below the yield stress of the material. Such materials tend to ‘clog up’ geometrical features like corners or side branches by forming rigid plugs, because the stress becomes insufficient to yield them. In this volume, Taylor-West and Hogg (J. Fluid Mech., vol. 941, 2022, A64) consider the problem of flow of an idealised viscoplastic fluid in a wedge forced by a disturbance far from the corner; this is the viscoplastic analogue of Moffatt's famous corner eddies. They demonstrate and describe the details of how the fluid clogs up the corner of the wedge with a rigid plug bounded by a viscoplastic eddy. More generally, this study provides a way to explain some of the details of cloggings in different geometries that have previously been observed in viscoplastic materials.
Evolution of convection in a layered porous medium
- Duncan R. Hewitt
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- Journal:
- Journal of Fluid Mechanics / Volume 941 / 25 June 2022
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- 06 May 2022, A56
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The effect of a series of thin, horizontal, low-permeability layers on convective motion from a distributed dense source along an upper boundary in an otherwise homogeneous, two-dimensional porous medium is considered. This set-up provides an idealised version of a relatively common form of heterogeneity in geological formations. The thickness and permeability of the thin layers are assumed to be small relative to the distance between them and the bulk permeability, respectively. As such, the layers can be parameterised by their impedance $\varOmega$ – a dimensionless ratio of the effective layer thickness and permeability – while the strength of convection is controlled by the dimensionless distance $H \gg 1$ between layers, which can also be interpreted as an effective Rayleigh number for the flow. The role of $\varOmega$ is explored with the aid of high-resolution numerical simulations, and simple analytical models are developed for the evolution of the mean concentration and the flux in the limits of small and large $\varOmega$. For intermediate values of $\varOmega$, the flow undergoes a transition from predominantly diffusive transfer across the layers to predominantly advective transfer, and the lateral scale of the flow can become very large. This transition is characterised and a simple model is developed.
Horizontal miscible displacements through porous media: the interplay between viscous fingering and gravity segregation
- Japinder S. Nijjer, Duncan R. Hewitt, Jerome A. Neufeld
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- Journal:
- Journal of Fluid Mechanics / Volume 935 / 25 March 2022
- Published online by Cambridge University Press:
- 26 January 2022, A14
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We consider miscible displacements in two-dimensional homogeneous porous media where the displacing fluid is less viscous and has a different density than the displaced fluid. We find that the dynamics evolve through nine possible regimes depending on the viscosity ratio, strength of density variations and the strength of the background flow, as characterized by the Péclet number. At early times the interface is dominated by longitudinal diffusion before undergoing a transition to a slumping regime where vertical flow is important. At intermediate times, vertical flow and diffusion can be neglected and there are three different limiting solutions: a fingering limit; an injection-driven gravity-current limit; and a density-driven gravity-current limit. Finally at late times, transverse diffusion becomes important and there is a transition from an apparent shutdown regime to a viscously enhanced Taylor-slumping regime. In each of the regimes, the dominant scalings are identified and reduced-order models for the evolution of the concentration field are developed. Lastly, three case studies are considered to illustrate the dominant physical balances in the geophysically relevant setting of geological $\textrm {CO}_2$ storage.
Water flow through sediments and at the ice-sediment interface beneath Sermeq Kujalleq (Store Glacier), Greenland
- Samuel H. Doyle, Bryn Hubbard, Poul Christoffersen, Robert Law, Duncan R. Hewitt, Jerome A. Neufeld, Charlotte M. Schoonman, Thomas R. Chudley, Marion Bougamont
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- Journal:
- Journal of Glaciology / Volume 68 / Issue 270 / August 2022
- Published online by Cambridge University Press:
- 08 December 2021, pp. 665-684
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Subglacial hydrology modulates basal motion but remains poorly constrained, particularly for soft-bedded Greenlandic outlet glaciers. Here, we report detailed measurements of the response of subglacial water pressure to the connection and drainage of adjacent water-filled boreholes drilled through kilometre-thick ice on Sermeq Kujalleq (Store Glacier). These measurements provide evidence for gap opening at the ice-sediment interface, Darcian flow through the sediment layer, and the forcing of water pressure in hydraulically-isolated cavities by stress transfer. We observed a small pressure drop followed by a large pressure rise in response to the connection of an adjacent borehole, consistent with the propagation of a flexural wave within the ice and underlying deformable sediment. We interpret the delayed pressure rise as evidence of no pre-existing conduit and the progressive decrease in hydraulic transmissivity as the closure of a narrow (< 1.5 mm) gap opened at the ice-sediment interface, and a reversion to Darcian flow through the sediment layer with a hydraulic conductivity of ≤ 10−6 m s−1. We suggest that gap opening at the ice-sediment interface deserves further attention as it will occur naturally in response to the rapid pressurisation of water at the bed.
High-Rayleigh-number convection in porous–fluid layers
- Thomas Le Reun, Duncan R. Hewitt
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- Journal:
- Journal of Fluid Mechanics / Volume 920 / 10 August 2021
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- 14 June 2021, A35
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We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh–Bénard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy–Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection in both the porous and the unconfined fluid regions, while the porous flow still remains strongly confined and governed by Darcy's law. We demonstrate that the heat flux and average thermal structure of the system can be predicted using previous results of convection in individual fluid or porous layers. We revisit a controversy about the role of subcritical ‘penetrative convection’ in the porous medium, and confirm that such induced flow does not contribute to the heat flux through the system. Lastly, we briefly study the temporal coupling between the two layers and find that the turbulent fluid convection above acts as a low-pass filter on the longer time-scale variability of convection in the porous layer.
Buoyancy-driven plumes in a layered porous medium
- Duncan R. Hewitt, Gunnar G. Peng, John R. Lister
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- Journal:
- Journal of Fluid Mechanics / Volume 883 / 25 January 2020
- Published online by Cambridge University Press:
- 26 November 2019, A37
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Thin, roughly horizontal low-permeability layers are a common form of large-scale heterogeneity in geological porous formations. In this paper, the dynamics of a buoyancy-driven plume in a two-dimensional layered porous medium is studied theoretically, with the aid of high-resolution numerical simulations. The medium is uniform apart from a thin, horizontal layer of a much lower permeability, located a dimensionless distance $L\gg 1$ below the dense plume source. If the dimensionless thickness $2\unicode[STIX]{x1D700}L$ and permeability $\unicode[STIX]{x1D6F1}$ of the low-permeability layer are small, the effect of the layer is found to be well parameterized by its impedance $\unicode[STIX]{x1D6FA}=2\unicode[STIX]{x1D700}L/\unicode[STIX]{x1D6F1}$. Five different regimes of flow are identified and characterized. For $\unicode[STIX]{x1D6FA}\ll L^{1/3}$, the layer has no effect on the plume, but as $\unicode[STIX]{x1D6FA}$ is increased the plume widens and spreads over the layer as a gravity current. For still larger $\unicode[STIX]{x1D6FA}$, the flow becomes destabilized by convective instabilities both below and above the layer, until, for $\unicode[STIX]{x1D6FA}\gg L$, the spread of the plume is dominated by convective mixing and buoyancy is transported across the layer by diffusion alone. Analytical models for the spread of the plume over the layer in the various different regimes are presented.
The elastic Landau–Levich problem on a slope
- Katarzyna L. P. Warburton, Duncan R. Hewitt, Jerome A. Neufeld
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- Journal:
- Journal of Fluid Mechanics / Volume 883 / 25 January 2020
- Published online by Cambridge University Press:
- 26 November 2019, A40
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The elastic analogue of the Landau–Levich dip-coating problem, in which a plate is withdrawn from a bath of fluid on whose surface lies a thin elastic sheet, is analysed for angle of withdrawal $\unicode[STIX]{x1D703}$ to the horizontal. The flow is controlled by the elasticity number, $El$, which is a measure of the relative importance of viscous and bending stresses, and $\unicode[STIX]{x1D703}$. The leading-order solution for small $El$ is a steady profile in which the thickness of the film on the plate is found to vary as $El^{3/4}/(1-\cos \unicode[STIX]{x1D703})^{5/8}$. This prediction is confirmed in the limit $\unicode[STIX]{x1D703}\ll 1$ by comparison with numerical simulation. Finally, the circumstances under which the assumption of a steady solution is no longer valid are discussed, and the time-dependent solution is described.
Stable and unstable miscible displacements in layered porous media
- Japinder S. Nijjer, Duncan R. Hewitt, Jerome A. Neufeld
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- Journal:
- Journal of Fluid Mechanics / Volume 869 / 25 June 2019
- Published online by Cambridge University Press:
- 25 April 2019, pp. 468-499
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The effect of permeability heterogeneities and viscosity variations on miscible displacement processes in porous media is examined using high-resolution numerical simulations and reduced theoretical modelling. The planar injection of one fluid into a fluid-saturated, two-dimensional porous medium with a permeability that varies perpendicular to the flow direction is studied. Three cases are considered, in which the injected fluid is equally viscous, more viscous or less viscous than the ambient fluid. In general it is found that the flow in each case evolves through three regimes. At early times, the flow exhibits the concentration evolves diffusively, independent of both the permeability structure and the viscosity ratio. At intermediate times, the flow exhibits different dynamics including channelling and fingering, depending on whether the injected fluid is more or less viscous than the ambient fluid, and depending on the relative magnitude of the viscosity and permeability variations. Finally, at late times, the flow becomes independent of the viscosity ratio and dominated by shear-enhanced (Taylor) dispersion. For each of the regimes identified above, we develop reduced-order models for the evolution of the transversely averaged concentration and compare them to the full numerical simulations.
The influence of a poroelastic till on rapid subglacial flooding and cavity formation
- Duncan R. Hewitt, Gregory P. Chini, Jerome A. Neufeld
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- Journal:
- Journal of Fluid Mechanics / Volume 855 / 25 November 2018
- Published online by Cambridge University Press:
- 24 September 2018, pp. 1170-1207
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We develop a model of the rapid propagation of water at the contact between elastic glacial ice and a poroelastic subglacial till, motivated by observations of the rapid drainage of supraglacial lakes in Greenland. By treating the ice as an elastic bending beam, the fluid dynamics of contact with the subglacial hydrological network, which is modelled as a saturated poroelastic till, can be examined in detail. The model describes the formation and dynamics of an axisymmetric subglacial cavity, and the spread of pore pressure, in response to injection of fluid. A combination of numerical simulation and asymptotic analysis is used to describe these dynamics for both a rigid and a deformable porous till, and for both laminar and turbulent fluid flow. For constant injection rates and laminar flow, the cavity is isostatic and its spread is controlled by bending of the ice and suction of pore water in the vicinity of the ice–till contact. For a deformable till, this control can be modified: generically, a flexural wave that is initially trapped in advance of the contact point relaxes over time by diffusion of pore pressure ahead of the cavity. While the dynamics are found to be relatively insensitive to the properties of the subglacial till during injection with a constant flux, significant dependence on the till properties is manifest during the subsequent spread of a constant volume. A simple hybrid turbulent–laminar model is presented to account for fast injection rates of water: in this case, self-similar turbulent propagation can initially control the spread of the cavity, but there is a transition to laminar control in the vicinity of the ice–till contact point as the flow slows. Finally, the model results are compared with recent geophysical observations of the rapid drainage of supraglacial lakes in Greenland; the comparison provides qualitative agreement and raises suggestions for future quantitative comparison.
The dynamics of miscible viscous fingering from onset to shutdown
- Japinder S. Nijjer, Duncan R. Hewitt, Jerome A. Neufeld
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- Journal of Fluid Mechanics / Volume 837 / 25 February 2018
- Published online by Cambridge University Press:
- 05 January 2018, pp. 520-545
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We examine the full ‘life cycle’ of miscible viscous fingering from onset to shutdown with the aid of high-resolution numerical simulations. We study the injection of one fluid into a planar two-dimensional porous medium containing another, more viscous fluid. We find that the dynamics are distinguished by three regimes: an early-time linearly unstable regime, an intermediate-time nonlinear regime and a late-time single-finger exchange-flow regime. In the first regime, the flow can be linearly unstable to perturbations that grow exponentially. We identify, using linear stability theory and numerical simulations, a critical Péclet number below which the flow remains stable for all times. In the second regime, the flow is dominated by the nonlinear coalescence of fingers which form a mixing zone in which we observe that the convective mixing rate, characterized by a convective Nusselt number, exhibits power-law growth. In this second regime we derive a model for the transversely averaged concentration which shows good agreement with our numerical experiments and extends previous empirical models. Finally, we identify a new final exchange-flow regime in which a pair of counter-propagating diffusive fingers slow exponentially. We derive an analytic solution for this single-finger state which agrees well with numerical simulations. We demonstrate that the flow always evolves to this regime, irrespective of the viscosity ratio and Péclet number, in contrast to previous suggestions.
Stability of three-dimensional columnar convection in a porous medium
- Duncan R. Hewitt, John R. Lister
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- Journal:
- Journal of Fluid Mechanics / Volume 829 / 25 October 2017
- Published online by Cambridge University Press:
- 14 September 2017, pp. 89-111
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The stability of steady convective exchange flow with a rectangular planform in an unbounded three-dimensional porous medium is explored. The base flow comprises a balance between vertical advection with amplitude $A$ in interleaving rectangular columns with aspect ratio $\unicode[STIX]{x1D709}\leqslant 1$ and horizontal diffusion between the columns. Columnar flow with a square planform ($\unicode[STIX]{x1D709}=1$) is found to be weakly unstable to a large-scale perturbation of the background temperature gradient, irrespective of $A$, but to have no stronger instability on the scale of the columns. This result provides a stark contrast to two-dimensional columnar flow (Hewitt et al., J. Fluid Mech., vol. 737, 2013, pp. 205–231), which, as $A$ is increased, is increasingly unstable to a perturbation on the scale of the columnar wavelength. For rectangular planforms with $\unicode[STIX]{x1D709}<1$, a critical aspect ratio is identified, below which a perturbation on the scale of the columns is the fastest growing mode, as in two dimensions. Scalings for the growth rate and the structure of this mode are identified, and are explained by means of an asymptotic expansion in the limit $\unicode[STIX]{x1D709}\rightarrow 0$. The difference between the stabilities of two-dimensional and three-dimensional exchange flow provides a potential explanation for the apparent difference in dominant horizontal scale observed in direct numerical simulations of two-dimensional and three-dimensional statistically steady ‘Rayleigh–Darcy’ convection at high Rayleigh numbers.
Shallow, gravity-driven flow in a poro-elastic layer
- Duncan R. Hewitt, Jerome A. Neufeld, Neil J. Balmforth
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- Journal:
- Journal of Fluid Mechanics / Volume 778 / 10 September 2015
- Published online by Cambridge University Press:
- 05 August 2015, pp. 335-360
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By combining Biot’s theory of poro-elasticity with standard shallow-layer scalings, a theoretical model is developed to describe axisymmetric gravity-driven flow through a shallow deformable porous medium. Motivated in part by observations of surface uplift around $\text{CO}_{2}$ sequestration sites, the model is used to explore the injection of a dense fluid into a horizontal, deformable porous layer that is initially saturated with another, less dense, fluid. The layer lies between a rigid base and a flexible overburden, both of which are impermeable. As the injected fluid spreads under gravity, the matrix deforms and the overburden lifts up. The coupled model predicts the location of the injected fluid as it spreads and the resulting uplift of the overburden due to deformation of the solid matrix. In general, the uplift spreads diffusively far ahead of the injected fluid. If fluid is injected with a constant flux and the medium is unbounded, both the uplift and the injected fluid spread in a self-similar fashion with the same similarity variable $\propto r/t^{1/2}$. The asymptotic form of this spreading is established. Results from a series of laboratory experiments, using polyacrylamide hydrogel particles to create a soft poro-elastic material, are compared qualitatively with the predictions of the model.
High Rayleigh number convection in a porous medium containing a thin low-permeability layer
- Duncan R. Hewitt, Jerome A. Neufeld, John R. Lister
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- Journal:
- Journal of Fluid Mechanics / Volume 756 / 10 October 2014
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- 05 September 2014, pp. 844-869
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Porous geological formations are commonly interspersed with thin, roughly horizontal, low-permeability layers. Statistically steady convection at high Rayleigh number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ra}$ is investigated numerically in a two-dimensional porous medium that is heated at the lower boundary and cooled at the upper, and contains a thin, horizontal, low-permeability interior layer. In the limit that both the dimensionless thickness $h$ and permeability $\Pi $ of the low-permeability layer are small, the flow is described solely by the impedance of the layer $\Omega = h/\Pi $ and by $\mathit{Ra}$. In the limit $\Omega \to 0$ (i.e. $h \to 0$), the system reduces to a homogeneous Rayleigh–Darcy (porous Rayleigh–Bénard) cell. Two notable features are observed as $\Omega $ is increased: the dominant horizontal length scale of the flow increases; and the heat flux, as measured by the Nusselt number $\mathit{Nu}$, can increase. For larger values of $\Omega $, $\mathit{Nu}$ always decreases. The dependence of the flow on $\mathit{Ra}$ is explored, over the range $2500 \leqslant \mathit{Ra} \leqslant 2\times 10^4$. Simple one-dimensional models are developed to describe some of the observed features of the relationship $\mathit{Nu}(\Omega )$.
High Rayleigh number convection in a three-dimensional porous medium
- Duncan R. Hewitt, Jerome A. Neufeld, John R. Lister
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- Journal:
- Journal of Fluid Mechanics / Volume 748 / 10 June 2014
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- 09 May 2014, pp. 879-895
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High-resolution numerical simulations of statistically steady convection in a three-dimensional porous medium are presented for Rayleigh numbers $Ra \leqslant 2 \times 10^4$. Measurements of the Nusselt number $Nu$ in the range $1750 \leqslant Ra \leqslant 2 \times 10^4$ are well fitted by a relationship of the form $Nu = \alpha _3 Ra + \beta _3$, for $\alpha _3 = 9.6 \times 10^{-3}$ and $\beta _3 = 4.6$. This fit indicates that the classical linear scaling $Nu \sim Ra$ is attained, and that $Nu$ is asymptotically approximately $40\, \%$ larger than in two dimensions. The dynamical flow structure in the range $1750 \leqslant Ra \leqslant 2\times 10^4$ is analysed, and the interior of the flow is found to be increasingly well described as $Ra \to \infty $ by a heat-exchanger model, which describes steady interleaving columnar flow with horizontal wavenumber $k$ and a linear background temperature field. Measurements of the interior wavenumber are approximately fitted by $k\sim Ra^{0.52 \pm 0.05}$, which is distinguishably stronger than the two-dimensional scaling of $k\sim Ra^{0.4}$.
Stability of columnar convection in a porous medium
- Duncan R. Hewitt, Jerome A. Neufeld, John R. Lister
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- Journal:
- Journal of Fluid Mechanics / Volume 737 / 25 December 2013
- Published online by Cambridge University Press:
- 22 November 2013, pp. 205-231
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Convection in a porous medium at high Rayleigh number $\mathit{Ra}$ exhibits a striking quasisteady columnar structure with a well-defined and $\mathit{Ra}$-dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ‘heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber $k$ and amplitude $\widehat{A}$ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter $A= \widehat{A}\mathit{Ra}/ k$. A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of $A$. The growth rate of the most unstable mode scales with ${A}^{4/ 9} $ for $A\gg 1$, and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh–Bénard (Rayleigh–Darcy) cell at high $\mathit{Ra}$, and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers $k$ greater than $k\sim {\mathit{Ra}}^{5/ 14} $ as $\mathit{Ra}\rightarrow \infty $. This stability criterion is consistent with hitherto unexplained numerical measurements of $k$ in a Rayleigh–Darcy cell.
Thixotropic gravity currents
- Duncan R. Hewitt, Neil J. Balmforth
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- Journal:
- Journal of Fluid Mechanics / Volume 727 / 25 July 2013
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- 14 June 2013, pp. 56-82
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We present a model for thixotropic gravity currents flowing down an inclined plane that combines lubrication theory for shallow flow with a rheological constitutive law describing the degree of microscopic structure. The model is solved numerically for a finite volume of fluid in both two and three dimensions. The results illustrate the importance of the degree of initial ageing and the spatio-temporal variations of the microstructure during flow. The fluid does not flow unless the plane is inclined beyond a critical angle that depends on the ageing time. Above that critical angle and for relatively long ageing times, the fluid dramatically avalanches downslope, with the current becoming characterized by a structured horseshoe-shaped remnant of fluid at the back and a raised nose at the advancing front. The flow is prone to a weak interfacial instability that occurs along the border between structured and de-structured fluid. Experiments with bentonite clay show broadly similar phenomenological behaviour to that predicted by the model. Differences between the experiments and the model are discussed.
Convective shutdown in a porous medium at high Rayleigh number
- Duncan R. Hewitt, Jerome A. Neufeld, John R. Lister
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- Journal:
- Journal of Fluid Mechanics / Volume 719 / 25 March 2013
- Published online by Cambridge University Press:
- 19 February 2013, pp. 551-586
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Convection in a closed domain driven by a dense buoyancy source along the upper boundary soon starts to wane owing to the increase of the average interior density. In this paper, theoretical and numerical models are developed of the subsequent long period of shutdown of convection in a two-dimensional porous medium at high Rayleigh number $\mathit{Ra}$. The aims of this paper are twofold. Firstly, the relationship between this slowly evolving ‘one-sided’ shutdown system and the statistically steady ‘two-sided’ Rayleigh–Bénard (RB) cell is investigated. Numerical measurements of the Nusselt number $\mathit{Nu}$ from an RB cell (Hewitt et al., Phys. Rev. Lett., vol. 108, 2012, 224503) are very well described by the simple parametrization $\mathit{Nu}= 2. 75+ 0. 0069\mathit{Ra}$. This parametrization is used in theoretical box models of the one-sided shutdown system and found to give excellent agreement with high-resolution numerical simulations of this system. The dynamical structure of shutdown can also be accurately predicted by measurements from an RB cell. Results are presented for a general power-law equation of state. Secondly, these ideas are extended to model more complex physical systems, which comprise two fluid layers with an equation of state such that the solution that forms at the (moving) interface is more dense than either layer. The two fluids are either immiscible or miscible. Theoretical box models compare well with numerical simulations in the case of a flat interface between the fluids. Experimental results from a Hele-Shaw cell and numerical simulations both show that interfacial deformation can dramatically enhance the convective flux. The applicability of these results to the convective dissolution of geologically sequestered ${\mathrm{CO} }_{2} $ in a saline aquifer is discussed.
Contributors
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. Douglas Meeks, Monica Jyotsna Melanchthon, Ilie Melniciuc-Puica, Everett Mendoza, Raymond A. Mentzer, William W. Menzies, Ina Merdjanova, Franziska Metzger, Constant J. Mews, Marvin Meyer, Carol Meyers, Vasile Mihoc, Gunner Bjerg Mikkelsen, Maria Inêz de Castro Millen, Clyde Lee Miller, Bonnie J. Miller-McLemore, Alexander Mirkovic, Paul Misner, Nozomu Miyahira, R. W. L. Moberly, Gerald Moede, Aloo Osotsi Mojola, Sunanda Mongia, Rebeca Montemayor, James Moore, Roger E. Moore, Craig E. Morrison O.Carm, Jeffry H. Morrison, Keith Morrison, Wilson J. Moses, Tefetso Henry Mothibe, Mokgethi Motlhabi, Fulata Moyo, Henry Mugabe, Jesse Ndwiga Kanyua Mugambi, Peggy Mulambya-Kabonde, Robert Bruce Mullin, Pamela Mullins Reaves, Saskia Murk Jansen, Heleen L. Murre-Van den Berg, Augustine Musopole, Isaac M. T. Mwase, Philomena Mwaura, Cecilia Nahnfeldt, Anne Nasimiyu Wasike, Carmiña Navia Velasco, Thulani Ndlazi, Alexander Negrov, James B. Nelson, David G. Newcombe, Carol Newsom, Helen J. Nicholson, George W. E. Nickelsburg, Tatyana Nikolskaya, Damayanthi M. A. Niles, Bertil Nilsson, Nyambura Njoroge, Fidelis Nkomazana, Mary Beth Norton, Christian Nottmeier, Sonene Nyawo, Anthère Nzabatsinda, Edward T. Oakes, Gerald O'Collins, Daniel O'Connell, David W. Odell-Scott, Mercy Amba Oduyoye, Kathleen O'Grady, Oyeronke Olajubu, Thomas O'Loughlin, Dennis T. Olson, J. Steven O'Malley, Cephas N. Omenyo, Muriel Orevillo-Montenegro, César Augusto Ornellas Ramos, Agbonkhianmeghe E. Orobator, Kenan B. Osborne, Carolyn Osiek, Javier Otaola Montagne, Douglas F. Ottati, Anna May Say Pa, Irina Paert, Jerry G. Pankhurst, Aristotle Papanikolaou, Samuele F. Pardini, Stefano Parenti, Peter Paris, Sung Bae Park, Cristián G. Parker, Raquel Pastor, Joseph Pathrapankal, Daniel Patte, W. Brown Patterson, Clive Pearson, Keith F. Pecklers, Nancy Cardoso Pereira, David Horace Perkins, Pheme Perkins, Edward N. Peters, Rebecca Todd Peters, Bishop Yeznik Petrossian, Raymond Pfister, Peter C. Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. Stewart, Cynthia Stokes Brown, Ken Stone, Anne Stott, Elizabeth Stuart, Monya Stubbs, Marjorie Hewitt Suchocki, David Kwang-sun Suh, Scott W. Sunquist, Keith Suter, Douglas Sweeney, Charles H. Talbert, Shawqi N. Talia, Elsa Tamez, Joseph B. Tamney, Jonathan Y. Tan, Yak-Hwee Tan, Kathryn Tanner, Feiya Tao, Elizabeth S. Tapia, Aquiline Tarimo, Claire Taylor, Mark Lewis Taylor, Bishop Abba Samuel Wolde Tekestebirhan, Eugene TeSelle, M. Thomas Thangaraj, David R. Thomas, Andrew Thornley, Scott Thumma, Marcelo Timotheo da Costa, George E. “Tink” Tinker, Ola Tjørhom, Karen Jo Torjesen, Iain R. Torrance, Fernando Torres-Londoño, Archbishop Demetrios [Trakatellis], Marit Trelstad, Christine Trevett, Phyllis Trible, Johannes Tromp, Paul Turner, Robert G. Tuttle, Archbishop Desmond Tutu, Peter Tyler, Anders Tyrberg, Justin Ukpong, Javier Ulloa, Camillus Umoh, Kristi Upson-Saia, Martina Urban, Monica Uribe, Elochukwu Eugene Uzukwu, Richard Vaggione, Gabriel Vahanian, Paul Valliere, T. J. Van Bavel, Steven Vanderputten, Peter Van der Veer, Huub Van de Sandt, Louis Van Tongeren, Luke A. Veronis, Noel Villalba, Ramón Vinke, Tim Vivian, David Voas, Elena Volkova, Katharina von Kellenbach, Elina Vuola, Timothy Wadkins, Elaine M. Wainwright, Randi Jones Walker, Dewey D. Wallace, Jerry Walls, Michael J. Walsh, Philip Walters, Janet Walton, Jonathan L. Walton, Wang Xiaochao, Patricia A. Ward, David Harrington Watt, Herold D. Weiss, Laurence L. Welborn, Sharon D. Welch, Timothy Wengert, Traci C. West, Merold Westphal, David Wetherell, Barbara Wheeler, Carolinne White, Jean-Paul Wiest, Frans Wijsen, Terry L. Wilder, Felix Wilfred, Rebecca Wilkin, Daniel H. Williams, D. Newell Williams, Michael A. Williams, Vincent L. Wimbush, Gabriele Winkler, Anders Winroth, Lauri Emílio Wirth, James A. Wiseman, Ebba Witt-Brattström, Teofil Wojciechowski, John Wolffe, Kenman L. Wong, Wong Wai Ching, Linda Woodhead, Wendy M. Wright, Rose Wu, Keith E. Yandell, Gale A. Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
-
- Book:
- The Cambridge Dictionary of Christianity
- Published online:
- 05 August 2012
- Print publication:
- 20 September 2010, pp xi-xliv
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