Route to chaos in porous-medium thermal convection

Kimura, Shigeo; Schubert, G.; Straus, Joe M.

doi:10.1017/s0022112086000162

Cited by 111 publications

(69 citation statements)

References 27 publications

(25 reference statements)

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“…For Ra slightly greater than 400, instabilities within the upper and lower thermal boundary layers generate small-scale features that are advected around the cell by the large-scale rolls. In this moderate Ra parameter regime, 400 Ra 1300, the resulting flow exhibits a series of transitions between periodic and quasi-periodic roll motions, as discussed in considerable detail by Kimura et al (1986Kimura et al ( , 1987, Aidun & Steen (1987) and Graham & Steen (1992, 1994. However, the rolls do not completely lose coherence until Ra 1300.…”

Section: Introductionmentioning

confidence: 89%

Structure and stability of steady porous medium convection at large Rayleigh number

2015

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This version is available at https://strathprints.strath.ac.uk/52305/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. A systematic investigation of unstable steady-state solutions of the Darcy-OberbeckBoussinesq equations at large values of the Rayleigh number Ra is performed to gain insight into two-dimensional porous medium convection in domains of varying aspectratio L. The steady convective states are shown to transport less heat than the statistically steady 'turbulent' flow realised at the same parameter values: the Nusselt number N u ∼ Ra for turbulent porous medium convection, while N u ∼ Ra 0.6 for the maximum heat-transporting steady solutions. A key finding is that the lateral scale of the heat-fluxmaximising solutions shrinks roughly as L ∼ Ra −0.5 , reminiscent of the decrease of the mean inter-plume spacing observed in turbulent porous medium convection as the thermal forcing is increased. A spatial Floquet analysis is performed to investigate the linear stability of the fully nonlinear steady convective states, extending a recent study by Hewitt et al. (J. Fluid Mech. 737, 2013) by treating a base convective state -and secondary stability modes -that satisfy appropriate boundary conditions along plane parallel walls. As in that study, a bulk instability mode is found for sufficiently small aspect-ratio base states. However, the growth rate of this bulk mode is shown to be significantly reduced by the presence of the walls. Beyond a certain critical Ra-dependent aspect-ratio, the base state is most strongly unstable to a secondary mode that is localised near the heated and cooled walls. Direct numerical simulations, strategically initialised to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically-steady porous medium convection results from a balance between the competing effects of these two types of inst...

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Section: Introductionmentioning

confidence: 89%

Structure and stability of steady porous medium convection at large Rayleigh number

2015

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“…For Ra < Ra crit = 4π 2 , the system is stable, and there is no flow (Nield & Bejan 2006). For 4π 2 < Ra 1300, the flow is characterized by large-scale convective rolls, which undergo a series of bifurcations that perturb, but do not completely break down, the background flow (Kimura, Schubert & Strauss 1986;Graham & Steen 1994). However, above Ra ≈ 1300, the rolls are broken down by vigorous plume shedding from the boundary layers at the top and bottom of the domain.…”

Section: Introductionmentioning

confidence: 99%

Convective shutdown in a porous medium at high Rayleigh number

2013

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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 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 Nu from an RB cell (Hewitt et al., Phys. Rev. Lett., vol. 108, 2012, 224503) are very well described by the simple parametrization Nu = 2.75 + 0.0069Ra. 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 CO 2 in a saline aquifer is discussed.

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“…Validation of the code was accomplished by Ž comparison with published results on thermo chem-. Ž ical convection in porous media Kimura et al, 1986;Rosenberg and Spera, 1992;Oldenburg and . Pruess, 1995 . …”

Section: Mol Mechmentioning

confidence: 99%

The formation and evolution of layered structures in porous media: effects of porosity and mechanical dispersion

Schoofs

Trompert

Hansen

2000

Physics of the Earth and Planetary Interiors

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Horizontally layered structures can develop in porous or partially molten environments, such as hydrothermal systems, magmatic intrusions and the early Earth's mantle. The porosity f of these natural environments is typically small. Since dissolved chemical elements unlike heat cannot diffuse through the solid rocks, heat and solute influence the interstitial fluid density in a different manner: heat advects slower than solute through the liquid by the factor f, while diffusion of heat through the bulk porous medium is larger by the factor f y1 times the ratio between the thermal and chemical diffusivities. By performing numerical experiments in which a rigid low-porosity medium is heated from below, we have studied the formation and evolution of layers in an initially stably stratified liquid. Growth of a convective layer through convective entrainment, the formation of a stable density interface on top of the layer and destabilization of the next layer are intimately Ž . Ž . linked. By monitoring the heat solute fluxes, it is observed that the transport of heat solute across the interface changes Ž . from convective entrainment towards a regime in which transfer is purely diffusive dispersive . Because this transition occurs before the stage at which the lower layer arrives at the thermal equilibrium, we conclude that the layer growth stops when the density interface on top has grown sufficiently strong to keep the ascending plumes in the lower layer from convectively entraining more fluid from above. A simple balance between the most important forces, exerted on a fluid parcel in the lower layer, is proposed to determine this transition. This force balance also indicates whether a density interface keeps intact, migrates upwards or breaks down during the further evolution of the layered sequence. Finally, mechanical dispersion tends to increase transport of chemically dissolved elements across the density interface. Since this reduces the density difference between the two adjacent layers, the thickness of the lower layer increases. q

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Route to chaos in porous-medium thermal convection

Cited by 111 publications

References 27 publications

Structure and stability of steady porous medium convection at large Rayleigh number

Structure and stability of steady porous medium convection at large Rayleigh number

Convective shutdown in a porous medium at high Rayleigh number

The formation and evolution of layered structures in porous media: effects of porosity and mechanical dispersion

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