24 results
Ablation of sloping ice faces into polar seawater
- Mainak Mondal, Bishakhdatta Gayen, Ross W. Griffiths, Ross C. Kerr
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- Journal:
- Journal of Fluid Mechanics / Volume 863 / 25 March 2019
- Published online by Cambridge University Press:
- 28 January 2019, pp. 545-571
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The effects of the slope of an ice–seawater interface on the mechanisms and rate of ablation of the ice by natural convection are examined using turbulence-resolving simulations. Solutions are obtained for ice slopes $\unicode[STIX]{x1D703}=2^{\circ }{-}90^{\circ }$, at a fixed ambient salinity and temperature, chosen to represent common Antarctic ocean conditions. For laminar boundary layers the ablation rate decreases with height, whereas in the turbulent regime the ablation rate is found to be height independent. The simulated laminar ablation rates scale with $(\sin \unicode[STIX]{x1D703})^{1/4}$, whereas in the turbulent regime it follows a $(\sin \unicode[STIX]{x1D703})^{2/3}$ scaling, both consistent with the theoretical predictions developed here. The reduction in the ablation rate with shallower slopes arises as a result of the development of stable density stratification beneath the ice face, which reduces turbulent buoyancy fluxes to the ice. The turbulent kinetic energy budget of the flow shows that, for very steep slopes, both buoyancy and shear production are drivers of turbulence, whereas for shallower slopes shear production becomes the dominant mechanism for sustaining turbulence in the convective boundary layer.
Dissolution of a sloping solid surface by turbulent compositional convection
- Craig D. McConnochie, Ross C. Kerr
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- Journal:
- Journal of Fluid Mechanics / Volume 846 / 10 July 2018
- Published online by Cambridge University Press:
- 08 May 2018, pp. 563-577
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We examine the dissolution of a sloping solid surface driven by turbulent compositional convection. The scaling analysis presented by Kerr & McConnochie (J. Fluid Mech., vol. 765, 2015, pp. 211–228) for the dissolution of a vertical wall is extended to the case of a sloping wall. The model has no free parameters and no dependence on height. It predicts that while the interfacial temperature and interfacial composition are independent of the slope, the dissolution velocity is proportional to $\cos ^{2/3}\unicode[STIX]{x1D703}$ , where $\unicode[STIX]{x1D703}$ is the angle of the sloping surface to the vertical. The analysis is tested by comparing it with laboratory measurements of the ablation of a sloping ice wall in contact with salty water. We apply the model to make predictions of the turbulent convective dissolution of a sloping ice shelf in the polar oceans.
Enhanced ablation of a vertical ice wall due to an external freshwater plume
- Craig D. McConnochie, Ross C. Kerr
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- Journal:
- Journal of Fluid Mechanics / Volume 810 / 10 January 2017
- Published online by Cambridge University Press:
- 28 November 2016, pp. 429-447
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We investigate the effect of an external freshwater plume on the dissolution of a vertical ice wall in salty water using laboratory experiments. We measure the plume velocity, the ablation velocity of the ice and the temperature at the ice wall. The freshwater volume flux, $Q_{s}$, is varied between experiments to determine where the resultant wall plume transitions from being dominated by the distributed buoyancy flux due to dissolution of the ice, to being dominated by the initial buoyancy flux, $B_{s}$. We find that when $B_{s}$ is significantly larger than the distributed buoyancy flux from dissolution, the plume velocity is uniform with height and is proportional to $B_{s}^{1/3}$, the interface temperature is independent of $B_{s}$, and the ablation velocity increases with $B_{s}$.
Simulation of convection at a vertical ice face dissolving into saline water
- Bishakhdatta Gayen, Ross W. Griffiths, Ross C. Kerr
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- Journal:
- Journal of Fluid Mechanics / Volume 798 / 10 July 2016
- Published online by Cambridge University Press:
- 31 May 2016, pp. 284-298
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We investigate the convection and dissolution rate generated when a wall of ice dissolves into seawater under Antarctic Ocean conditions. In direct numerical simulations three coupled interface equations are used to solve for interface temperature, salinity and ablation velocity, along with the boundary layer flow and transport. The main focus is on ambient water temperatures between $-1\,^{\circ }\text{C}$ and $6\,^{\circ }\text{C}$ and salinities around 35 ‰, where diffusion of salt to the ice–water interface depresses the freezing point and enhances heat diffusion to the ice. We show that fluxes of both heat and salt to the interface are significant in governing the dissolution of ice, and the ablation velocity agrees well with experiments and a recent theoretical prediction. The same turbulent flow dynamics and ablation rate are expected to apply at any depth in a deeper ocean water column (after choosing the relevant pressure coefficient for the liquidus temperature). At Grashof numbers currently accessible by direct numerical simulation, turbulence is generated both directly from buoyancy flux and from shear production in the buoyancy-driven boundary layer flow, whereas shear production by the convective flow is expected to be more important at geophysical scales. The momentum balance in the boundary layer is dominated by buoyancy forcing and wall stress, with the latter characterised by a large drag coefficient.
The effect of a salinity gradient on the dissolution of a vertical ice face
- Craig D. McConnochie, Ross C. Kerr
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- Journal:
- Journal of Fluid Mechanics / Volume 791 / 25 March 2016
- Published online by Cambridge University Press:
- 24 February 2016, pp. 589-607
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We investigate experimentally the effect of stratification on a vertical ice face dissolving into cold salty water. We measure the interface temperature, ablation velocity and turbulent plume velocity over a range of salinity gradients and compare our measurements with results of similar experiments without a salinity gradient (Kerr & McConnochie, J. Fluid Mech., vol. 765, 2015, pp. 211–228; McConnochie & Kerr, J. Fluid Mech., vol. 787, 2016, pp. 237–253). We observe that stratification acts to reduce the ablation velocity, interface temperature, plume velocity and plume acceleration. We define a stratification parameter, $S=N^{2}Q/{\it\Phi}_{o}$, that describes where stratification will be important, where $N$ is the Brunt–Väisälä frequency, $Q$ is the height-dependent plume volume flux and ${\it\Phi}_{o}$ is the buoyancy flux per unit area without stratification. The relevance of this stratification parameter is supported by our experiments, which deviate from the homogeneous theory at approximately $S=1$. Finally, we calculate values for the stratification parameter at a number of ice shelves and conclude that ocean stratification will have a significant effect on the dissolution of both the Antarctic and Greenland ice sheets.
The turbulent wall plume from a vertically distributed source of buoyancy
- Craig D. McConnochie, Ross C. Kerr
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- Journal:
- Journal of Fluid Mechanics / Volume 787 / 25 January 2016
- Published online by Cambridge University Press:
- 15 December 2015, pp. 237-253
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We experimentally investigate the turbulent wall plume that forms next to a uniformly distributed source of buoyancy. Our experimental results are compared with the theoretical model and experiments of Cooper & Hunt (J. Fluid Mech., vol. 646, 2010, pp. 39–58). Our experiments give a top-hat entrainment coefficient of $0.048\pm 0.006$. We measure a maximum vertical plume velocity that follows the scaling predicted by Cooper & Hunt but is significantly smaller. Our measurements allow us to construct a turbulent plume model that predicts all plume properties at any height. We use this plume model to calculate plume widths, velocities and Reynolds numbers for typical dissolving icebergs and ice fronts and for a typical room with a heated or cooled vertical surface.
Dissolution of a vertical solid surface by turbulent compositional convection
- Ross C. Kerr, Craig D. McConnochie
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- Journal:
- Journal of Fluid Mechanics / Volume 765 / 25 February 2015
- Published online by Cambridge University Press:
- 19 January 2015, pp. 211-228
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We examine the dissolution of a vertical solid surface in the case where the heat and mass transfer is driven by turbulent compositional convection. A theoretical model of the turbulent dissolution of a vertical wall is developed, which builds on the scaling analysis presented by Kerr (J. Fluid Mech., vol. 280, 1994, pp. 287–302) for the turbulent dissolution of a horizontal floor or roof. The model has no free parameters and no dependence on height. The analysis is tested by comparing it with laboratory measurements of the ablation of a vertical ice wall in contact with salty water. The model is found to accurately predict the dissolution velocity for water temperatures up to approximately 5–$6\,^{\circ }\text{C}$, where there is a transition from turbulent dissolution to turbulent melting. We quantify the turbulent convective dissolution of vertical ice bodies in the polar oceans, and compare our results with some field observations.
Rayleigh–Taylor instability of an inclined buoyant viscous cylinder
- JOHN R. LISTER, ROSS C. KERR, NICK J. RUSSELL, ANDREW CROSBY
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- Journal:
- Journal of Fluid Mechanics / Volume 671 / 25 March 2011
- Published online by Cambridge University Press:
- 01 February 2011, pp. 313-338
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The Rayleigh–Taylor instability of an inclined buoyant cylinder of one very viscous fluid rising through another is examined through linear stability analysis, numerical simulation and experiment. The stability analysis represents linear eigenmodes of a given axial wavenumber as a Fourier series in the azimuthal direction, allowing the use of separable solutions to the Stokes equations in cylindrical polar coordinates. The most unstable wavenumber k∗ is long-wave if both the inclination angle α and the viscosity ratio λ (internal/external) are small; for this case, k∗ ∝ max{α, (λ ln λ−1)1/2} and thus a small angle in experiments can have a significant effect for λ ≪ 1. As α increases, the maximum growth rate decreases and the upward propagation rate of disturbances increases; all disturbances propagate without growth if the cylinder is sufficiently close to vertical, estimated as α ≳ 70°. Results from the linear stability analysis agree with numerical calculations for λ = 1 and experimental observations. A point-force numerical method is used to calculate the development of instability into a chain of individual plumes via a complex three-dimensional flow. Towed-source experiments show that nonlinear interactions between neighbouring plumes are important for α ≳ 20° and that disturbances can propagate out of the system without significant growth for α ≳ 40°.
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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. 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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
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- Book:
- The Cambridge Dictionary of Christianity
- Published online:
- 05 August 2012
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- 20 September 2010, pp xi-xliv
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The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface
- John R. Lister, Ross C. Kerr
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- Journal of Fluid Mechanics / Volume 203 / June 1989
- Published online by Cambridge University Press:
- 26 April 2006, pp. 215-249
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Viscous gravity currents resulting from the introduction of fluid between an upper layer of fluid of lesser density and a lower layer of greater density are analysed. The nonlinear equations governing the spread and shape of the intrusion are formulated for the cases of intrusion at low Reynolds number between deep ambient layers and of flow over a shallow layer of viscous fluid with a rigid lower boundary. Similarity solutions of these equations are obtained in both two-dimensional and axisymmetric geometries, under the assumption that the volume of intruding fluid increases with time like tα. The theoretical predictions are shown to be in reasonable agreement with experimental observations of the spreading of glucose syrups and of viscous hydrocarbons between fluid layers of differing densities. Scaling arguments are used to derive many new results for the rates of spread of intrusions in a wide variety of further situations. A compendium of spreading relations, including some previously isolated results, is derived within a coherent framework and tabulated.
The transient behaviour of alloys solidified from below prior to the formation of chimneys
- M. Grae Worster, Ross C. Kerr
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- Journal:
- Journal of Fluid Mechanics / Volume 269 / 25 June 1994
- Published online by Cambridge University Press:
- 26 April 2006, pp. 23-44
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We investigate interactions between interfacial disequilibrium and compositional convection during the freezing of an alloy from below to form a mushy layer. A theoretical model is developed in which a stagnant mushy layer underlies a melt that is convecting vigorously, driven by compositional gradients associated with undercooling at the mush-liquid interface. In a series of laboratory experiments, we measure the interfacial undercooling in aqueous solutions of ammonium chloride contaminated to varying degrees by copper sulphate. It has recently been found (Huppert & Hallworth 1993) that a small amount of copper sulphate added to a solution of ammonium chloride significantly inhibits the formation of chimneys in the mushy layer that forms when the solution is cooled below its liquidus. It is our thesis that this phenomenon can be explained in large part by the consequences of the interactions between compositional convection and interfacial undercooling that are investigated herein. The measured undercooling is a function of the rate of advance of the interface and is found to be a very strong, increasing function of the concentration of copper sulphate in solution. The theoretical model is evaluated using parameter values appropriate to the experimental system and it is found that the transient development of the mushy layer depends significantly on the level of interfacial disequilibrium. In particular, it is predicted that the time taken for the Rayleigh number associated with the mushy layer to reach any particular value increases enormously as the level of interfacial disequilibrium increases and that the Rayleigh number can have an upper bound that is less than the critical value needed for the onset of convection within the mushy layer. This suggests that the formation of chimneys in the mushy layer can be similarly delayed or prohibited, in agreement with the experimental findings of Huppert & Hallworth (1993). Additionally, the model predicts that under certain conditions the solid fraction can increase away from the cooled boundary leading to trapping of the interstitial liquid. The model also describes a mechanism for macrosegregation of alloys cooled and solidified from below.
Solidification of an alloy cooled from above Part 1. Equilibrium growth
- Ross C. Kerr, Andrew W. Woods, M. Grae Worster, Herbert E. Huppert
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- Journal:
- Journal of Fluid Mechanics / Volume 216 / July 1990
- Published online by Cambridge University Press:
- 26 April 2006, pp. 323-342
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The interaction between the solidification and convection that occurs when a melt is cooled from above is investigated in a series of three papers. In these papers we consider a two-component melt that partially solidifies to leave a buoyant residual fluid. The solid forms a mushy layer of dendritic crystals, the interstices of which accommodate the residual fluid. The heat extraction through the upper boundary, necessary to promote solidification, drives convection at high Rayleigh numbers in the melt below the mushy layer. The convection enhances the heat transfer from the melt and alters the rate of solidification. In this paper the various phenomena are studied in a series of laboratory experiments in which ice is frozen from aqueous solutions of isopropanol. The experiments are complemented by the development of a general theoretical model in which the mush is treated as a continuum phase with thermodynamic properties that are functions of the local solid fraction. The model, which is based upon principles of equilibrium thermodynamics and local conservation of heat and solute, produces results in good agreement with the experimental data. Careful comparisons between this theory and experiments suggest the need to explore non-equilibrium effects, which are investigated in Parts 2 and 3.
Convection and particle entrainment driven by differential sedimentation
- Herbert E. Huppert, Ross C. Kerr, John R. Lister, J. Stewart Turner
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- Journal of Fluid Mechanics / Volume 226 / May 1991
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- 26 April 2006, pp. 349-369
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When a suspension of small particles is overlain by a clear fluid whose density is greater than that of the interstitial fluid, but less than that of the bulk suspension, the settling of the dense suspended particles can lead to vigorous convection in the overlying fluid. This novel situation is investigated experimentally and theoretically. A sharp interface is observed between the convecting upper region and a stagnant lower region in which there is unimpeded sedimentation at low Reynolds number. There is no transport of fluid from the upper region into the lower, though there is mixing of both buoyant fluid and entrained particles from the lower region into the upper. The interface between the two regions is found to descend at a constant velocity. Systematic laboratory measurements have determined how this velocity depends on the densities of the layers and the distributions of settling velocities of the particles. A theoretical description is developed which calculates the evolution of the density of the lower region due to differential sedimentation of polydisperse particles. Buoyancy arguments based on the calculated density profile are used to place upper and lower bounds on the amount of particle entrainment into the upper layer and on the rate of fall of the interface between the convecting and sedimenting regions. The theoretical predictions are in good agreement with the experimental observations. The analysis of the interaction between convection and sedimentation in the system considered here may be particularly relevant to the description of evolving crystal-rich layers in magma chambers and of silt-laden outflow from rivers, and has a wide range of other industrial, environmental and geological applications.
The effect of geometry on the gravitational instability of a buoyant region of viscous fluid
- John R. Lister, Ross C. Kerr
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- Journal of Fluid Mechanics / Volume 202 / May 1989
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- 26 April 2006, pp. 577-594
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The low-Reynolds-number stability of a region of buoyant fluid surrounded by denser fluid is analysed in two situations. In the first study, the buoyant fluid lies in a thin layer sandwiched between two denser and much deeper layers. The growth rate and wavelength of the most unstable sinusoidal perturbation are calculated and the effects of the viscosity ratios and density differences between the fluids are investigated. It is found that if the buoyant fluid is much less viscous than the overlying fluid then, in quite general circumstances, both the most unstable wavelength and the corresponding growth rate are inversely proportional to the cube root of the viscosity of the buoyant fluid. A physical explanation of this result is given by scaling analysis of the total dissipation. In the second study, the buoyant fluid takes the form of a cylinder rising through a uniform environment. The eigenmodes of small perturbation about this state of motion are found for each axial wavenumber in terms of Fourier series of separable solutions to the Stokes equations. In contrast to the first study, it is found that the most unstable wavelength and growth rate are asymptotically independent of the viscosity of the buoyant fluid when this viscosity is small.
The difference between the results of the two studies is of importance, particularly for geophysical applications in which viscosity ratios are very large. Previous models of linear regions of volcanism at mid-ocean ridges and at island arcs have assumed that results obtained in simple two-layered systems can be generalized to other geometries. The conclusions of these models are discussed in the light of the stability results for a cylindrical (and hence linea.
Dissolving driven by vigorous compositional convection
- Ross C. Kerr
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- Journal of Fluid Mechanics / Volume 280 / 10 December 1994
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- 26 April 2006, pp. 287-302
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The one-dimensional dissolution that occurs when a binary melt is placed above or below a solid of a different composition is examined both theoretically and experimentally. In the case considered, the dissolution is driven by vigorous compositional convection that results from a Rayleigh-Bénard instability of the compositional boundary layer in the vicinity of the dissolving solid. A scaling analysis is used to derive theoretical expressions for both the dissolving velocity and the interfacial fluid concentration. Laboratory experiments are also described in which ice is dissolved when it is overlain or underlain by aqueous solutions. The measured dissolving velocities are consistent with the theoretical expressions, and yield estimates of the critical Rayleigh number for boundary-layer instability. The results of this study are then applied to predict the rate at which dissolution will occur when undersaturated mixed magmas are generated during the periodic replenishment of large basaltic magma chambers in the Earth's crust.
Melting driven by vigorous compositional convection
- Ross C. Kerr
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- Journal of Fluid Mechanics / Volume 280 / 10 December 1994
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- 26 April 2006, pp. 255-285
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The melting of a solid in contact with a hot fluid is quantified for the case in which a difference between the densities of the fluid and of the melted solid is able to drive vigorous compositional convection. A scaling analysis is first used to obtain a theoretical expression for the melting rate that is valid for a certain range of Stefan numbers. This expression is then compared with melting velocities measured in laboratory experiments in which ice and wax are melted when they are overlain or underlain by hot aqueous solutions. The melting velocities are consistent with the theoretical expression, and are found to depend on the heats of solution that are released when the melted solids mix with the solutions. The experiments also indicate that, for vigorous convection to occur during the melting of a floor, the unstable compositional buoyancy needs to be at least twice the stabilizing thermal buoyancy.
An important geological situation in which melting occurs is when large volumes of basaltic magma are intruded into the Earth's continental crust. The theoretical and experimental results are used and extended to examine quantitatively the melting of the floor and walls of the magma chamber, and of crustal blocks that fall into the chamber.
Further results for convection driven by the differential sedimentation of particles
- Ross C. Kerr, John R. Lister
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- Journal of Fluid Mechanics / Volume 243 / October 1992
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- 26 April 2006, pp. 227-245
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When a well-mixed suspension of small particles is emplaced below a clear fluid whose density is greater than that of the interstitial fluid, but less than that of the bulk suspension, the subsequent settling of the dense particles releases buoyant interstitial fluid and drives convection in the overlying layer. Mixing of interstitial fluid and some entrained particles into the overlying fluid causes the density of the overlying fluid to evolve with time and changes the rate of descent of the interface between the sedimenting and convecting regions. These effects are investigated experimentally in a simple rectangular geometry using suspensions of spherical glass particles. It is found that the convecting region is well mixed in both composition and particle concentration and that the interfacial velocity may be predicted from the instantaneous (uniform) bulk density of the upper layer and the distribution of the particle settling velocities. In the case of an overlying density gradient, the convection does not extend through the depth of the overlying fluid but erodes the base of the gradient to form a well-mixed layer between the gradient and the sedimenting fluid. On completion of the first cycle of sedimentation-driven convection, sedimentation from this well-mixed layer produces further cycles of sedimentation-driven convection, which are of successively decreasing intensity and increasing duration. Whether the overlying fluid is uniform or stratified, both theory and experiment show that the particles that are lifted into the convection are smaller on average than those which settle at the base of the lower layer. Thus, when the lifted particles are eventually allowed to settle there is a discontinuity generated in the variation of the size distribution of particles with height in the final sedimented pile. This phenomenon may be an important mechanism for secondary layering in the deposits from turbidity currents and pyroclastic flows.
Solidification of an alloy cooled from above. Part 3. Compositional stratification within the solid
- Ross C. Kerr, Andrew W. Woods, M. Grae Worster, Herbert E. Huppert
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- Journal of Fluid Mechanics / Volume 218 / September 1990
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- 26 April 2006, pp. 337-354
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This is the third of a series of papers which investigates the evolution of a binary alloy that is cooled from above and releases buoyant residual fluid as one component of the alloy is preferentially incorporated within the solid. This paper focuses on the compositional zonation that is produced when the melt is completely solidified. Parts 1 and 2 considered the temperature of the cooled boundary to be greater than the eutectic temperature of the alloy so that only partial solidification of the alloy could occur. Here we extend the study by investigating the effects that arise when the cooling temperature is less than the eutectic temperature. The formation of a completely solid layer results, which extends from the cooling plate down to a mushy zone of dendritic crystals and interstitial melt. The melt below this mushy layer is convectively unstable because it is cooled from above. This generates vigorous thermal convection. Eventually the melt becomes completely solidified and a compositionally zoned solid is formed. The solid below the cooling plate is shown to be of fixed bulk composition until it reaches the depth that the interface between the mushy layer and melt occupied when the melt first became saturated. Below this level, the composition of the solid decreases with depth until it merges into the solid growing from the floor, whose composition is nearly equal to that of the heavier component of the alloy. Results of laboratory experiments with aqueous solutions of sodium sulphate are in good agreement with our quantitative predictions. The model is used to give an interpretation of profiles of the composition of magnesium oxide found within solidified komatiite lavas.
Solidification of an alloy cooled from above Part 2. Non-equilibrium interfacial kinetics
- Ross C. Kerr, Andrew W. Woods, M. Grae Worster, Herbert E. Huppert
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- Journal:
- Journal of Fluid Mechanics / Volume 217 / August 1990
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- 26 April 2006, pp. 331-348
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The model developed in Part 1 for the solidification and convection that occurs when an alloy is cooled from above is extended to investigate the role of disequilibrium at the mush–liquid interface. Small departures from equilibrium are important because in a convecting system an interfacial temperature below its equilibrium value can drive the bulk temperature of the melt below its liquidus. This behaviour is observed in experiments and can result in crystallization within and at the base of the convecting melt. The additional crystals formed in the interior can settle to the base of the fluid and continue to grow, causing the composition of the melt to change. This ultimately affects the solidification at the roof. The effects of disequilibrium are explored in this paper by replacing the condition of marginal equilibrium at the interface used in the model of Part 1 with a kinetic growth law of the form $\dot{h}_1 = {\cal G}\delta T$, where $\dot{h}_1$ is the rate of advance of the mush–liquid interface, δT is the amount by which the interfacial temperature is below the liquidus temperature of the melt and [Gscr ] is an empirical constant. This modification enables the model to predict very accurately both the growth of the mushy layer and the development of supersaturation in the isopropanol experiments described in Part 1. An additional series of experiments, using aqueous solutions of sodium sulphate, is presented in which the development of supersaturation leads to the internal nucleation and growth of crystals. A further extension of the model is introduced which successfully accounts for this internal crystal growth and the changing composition of the melt. We discuss the implications of this work for geologists studying the formation of igneous rocks. Important conclusions include the facts that cooling the roof of a magma chamber can lead to crystallization at its floor and that vigorous convection can occur in a magma chamber even when there is no initial superheat.
Patterns of solidification in channel flows with surface cooling
- ROSS W. GRIFFITHS, ROSS C. KERR, KATHARINE V. CASHMAN
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- Journal of Fluid Mechanics / Volume 496 / 10 December 2003
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- 01 December 2003, pp. 33-62
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Understanding the rates of cooling and solidification in laminar flows down sloping channels is central to predicting the advance of lava flows. The mechanisms involved include thermal convection and a competition between shear strain rate and the rate of formation of solid at the chilled surface of the flow. We report experiments in which polyethylene glycol wax flows in a laminar fashion down an inclined, open channel of rectangular cross-section under cold water. Two distinctly different flow regimes are recognized: ‘tube’ flow in which solidification of the flow surface creates a stationary roof while melt continues to flow through a relatively well-insulated ‘tube’ beneath, and a ‘mobile crust’ regime in which a solid surface crust develops only in the centre of the channel. In the latter regime the crust is carried down the channel, separated from the walls by crust-free shear regions in which cooling produces only dispersed fragments of solid owing to the effects of shearing. This flow structure is quasi-invariant over a large distance downstream. We show that thermal convection takes place in organized rolls that have axes aligned with the shear flow, and conclude that transition between the two flow regimes occurs at a critical value of the combined parameter $\vartheta \,{=}\, \psi (\hbox{\it Ra}{/}R_0)^{1/3}$, where $\psi \,{=}\, U_0 t_s{/}H_0$ is the ratio of a surface solidification timescale $t_{s}$ to a shearing timescale $H_0 {/} U_0 $, $H_{0}$ and $U_{0}$ are the flow depth and centreline surface velocity in the absence of solidification, {\it Ra} is a Rayleigh number and $R_{0}$ is a constant.