Solutal convection in porous media is thought to be controlled by the molecular Rayleigh number, Ra m , the ratio of the buoyant driving force over diffusive dissipation. The mass flux should increase linearly with Ra m and the finger spacing should decrease as Ra −1∕2 m. Instead, our experiments find that flux levels off at large Ra m and finger spacing increases with Ra m . Here we show that the convective pattern is controlled by a dispersive Rayleigh number, Ra d , balancing buoyancy and dispersion. Increasing the bead size of the porous medium increases Ra m but decreases Ra d and hence coarsens the pattern. While the flux is predominantly controlled by Ra m , the anisotropy of mechanical dispersion leads to an asymmetry in the pattern that limits the flux at large bead sizes.Plain Language Summary Pattern formation in simple physical systems is intriguing, and convection in porous media is an example that was thought to be well understood. Convection is controlled by the balance between buoyant driving forces and dissipative mechanisms such as diffusion that smear out the concentration and hence density differences. Here we use simple laboratory experiments to show that the convective pattern is controlled by a different process than previously thought. A Rayleigh number based on mechanical dispersion, which is independent of fluid properties, predicts the flow pattern of solutal convection in bead packs.

A new computational procedure for numerically solving a class of variational problems arising from rigorous upper bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Bénard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number Nu, as a function of the Rayleigh number Ra, Prandtl number Pr, and domain aspect ratio L in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents α and β in the presumed Nu ∼ Pr α Ra β scaling relation. The computations clearly show that for Ra ≤ 10 10 at fixed L = 2 √ 2, Nu ≤ 0.106Pr 0 Ra 5/12 , which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra

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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Motivated by geological carbon dioxide (CO 2 ) storage, many recent studies have investigated the fluid dynamics of solutal convection in porous media. Here we study the convective dissolution of CO 2 in a closed system, where the pressure in the gas declines as convection proceeds. This introduces a negative feedback that reduces the convective dissolution rate even before the brine becomes saturated. We analyse the case of an ideal gas with a solubility given by Henry's law, in the limits of very low and very high Rayleigh numbers. The equilibrium state in this system is determined by the dimensionless dissolution capacity, Π, which gives the fraction of the gas that can be dissolved into the underlying brine. Analytic approximations of the pure diffusion problem with Π > 0, show that the diffusive base state is no longer self-similar and that diffusive mass transfer declines rapidly with time. Direct numerical simulations at high Rayleigh numbers show that no constant flux regime exists for Π > 0; nevertheless, the quantity F/C 2 s remains constant, where F is the dissolution flux and C s is the dissolved concentration at the top of the domain. Simple mathematical models are developed to predict the evolution of C s and F for high-Rayleigh-number convection in a closed system. The negative feedback that limits convection in closed systems may explain the persistence of natural CO 2 accumulations over millennial timescales.

High-Rayleigh-number ($Ra$) convection in an inclined two-dimensional porous layer is investigated using direct numerical simulations (DNS) and stability and variational upper-bound analyses. When the inclination angle $\unicode[STIX]{x1D719}$ of the layer satisfies $0^{\circ }<\unicode[STIX]{x1D719}\lesssim 25^{\circ }$, DNS confirm that the flow exhibits a three-region wall-normal asymptotic structure in accord with the strictly horizontal ($\unicode[STIX]{x1D719}=0^{\circ }$) case, except that as $\unicode[STIX]{x1D719}$ is increased the time-mean spacing between neighbouring interior plumes also increases substantially. Both DNS and upper-bound analysis indicate that the heat transport enhancement factor (i.e. the Nusselt number) $Nu\sim CRa$ with a $\unicode[STIX]{x1D719}$-dependent prefactor $C$. When $\unicode[STIX]{x1D719}>\unicode[STIX]{x1D719}_{t}$, however, where $30^{\circ }<\unicode[STIX]{x1D719}_{t}<32^{\circ }$ independently of $Ra$, the columnar flow structure is completely broken down: the flow transitions to a large-scale travelling-wave convective roll state, and the heat transport is significantly reduced. To better understand the physics of inclined porous medium convection at large $Ra$ and modest inclination angles, a spatial Floquet analysis is performed, yielding predictions of the linear stability of numerically computed, fully nonlinear steady convective states. The results show that there exist two types of instability when $\unicode[STIX]{x1D719}\neq 0^{\circ }$: a bulk-mode instability and a wall-mode instability, consistent with previous findings for $\unicode[STIX]{x1D719}=0^{\circ }$ (Wen et al., J. Fluid Mech., vol. 772, 2015, pp. 197–224). The background flow induced by the inclination of the layer intensifies the bulk-mode instability during its subsequent nonlinear evolution, thereby favouring increased spacing between the interior plumes relative to that observed in convection in a horizontal porous layer.

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