## Doctoral Dissertations

Fall 2015

Dissertation

#### Program or Major

Applied Mathematics

#### Degree Name

Doctor of Philosophy

Gregory P. Chini

John P. McHugh

Christopher M. White

#### Abstract

Buoyancy-driven convection in fluid-saturated porous media is a key environmental and technological process, with applications ranging from carbon dioxide storage in terrestrial aquifers to the design of compact heat exchangers. Porous medium convection is also a paradigm for forced-dissipative infinite-dimensional dynamical systems, exhibiting spatiotemporally chaotic dynamics if not true" turbulence. The objective of this dissertation research is to quantitatively characterize the dynamics and heat transport in two-dimensional horizontal and inclined porous medium convection between isothermal plane parallel boundaries at asymptotically large values of the Rayleigh number $Ra$ by investigating the emergent, quasi-coherent flow. This investigation employs a complement of direct numerical simulations (DNS), secondary stability and dynamical systems theory, and variational analysis.

The DNS confirm the remarkable tendency for the interior flow to self-organize into closely-spaced columnar plumes at sufficiently large $Ra$ (up to $Ra \simeq 10^5$), with more complex spatiotemporal features being confined to boundary layers near the heated and cooled walls. The relatively simple form of the interior flow motivates investigation of unstable steady and time-periodic convective states at large $Ra$ as a function of the domain aspect ratio $L$. To gain insight into the development of spatiotemporally chaotic convection, the (secondary) stability of these fully nonlinear states to small-amplitude disturbances is investigated using a spatial Floquet analysis. The results indicate that there exist two distinct modes of instability at large $Ra$: a bulk instability mode and a wall instability mode. The former usually is excited by long-wavelength disturbances and is generally much weaker than the latter. DNS, strategically initialized 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 an interplay between the competing effects of these two types of instability.

Upper bound analysis is then employed to investigate the dependence of the heat transport enhancement factor, i.e. the Nusselt number $Nu$, on $Ra$ and $L$. To solve the optimization problems arising from the background field" upper-bound variational analysis, a novel two-step algorithm in which time is introduced into the formulation is developed. The new algorithm obviates the need for numerical continuation, thereby enabling the best available bounds to be computed up to $Ra\approx 2.65\times 10^4$. A mathematical proof is given to demonstrate that the only steady state to which this numerical algorithm can converge is the required global optimal of the variational problem. Using this algorithm, the dependence of the bounds on $L(Ra)$ is explored, and a minimal flow unit" is identified. Finally, the upper bound variational methodology is also shown to yield quantitatively useful predictions of $Nu$ and to furnish a functional basis that is naturally adapted to the boundary layer dynamics at large $Ra$.

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