19 research outputs found
POROUS MEDIUM CONVECTION AT LARGE RAYLEIGH NUMBER: STUDIES OF COHERENT STRUCTURE, TRANSPORT, AND REDUCED DYNAMICS
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 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 (up to ), 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 as a function of the domain aspect ratio . 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 : 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 , on and . 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 . 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 is explored, and a ``minimal flow unit is identified. Finally, the upper bound variational methodology is also shown to yield quantitatively useful predictions of and to furnish a functional basis that is naturally adapted to the boundary layer dynamics at large
High-wavenumber steady solutions of two-dimensional Rayleigh--B\'enard convection between stress-free boundaries
Recent investigations show that steady solutions share many features with
turbulent Rayleigh--B\'enard convection (RBC) and form the state space skeleton
of turbulent dynamics. Previous computations of steady roll solutions in
two-dimensional (2D) RBC between no-slip boundaries reveal that for fixed
Rayleigh number and Prandtl number , the heat-flux-maximizing solution
is always in the high-wavenumber regime. In this study, we explore the
high-wavenumber steady convection roll solutions that bifurcate supercritically
from the motionless conductive state for 2D RBC between stress-free boundaries.
Our computations confirm the existence of a local heat-flux-maximizing solution
in the high-wavenumber regime. To elucidate the asymptotic properties of this
solution, we perform computations over eight orders of magnitude in the
Rayleigh number, , and two orders of magnitude in
the Prandtl number, . The numerical results
indicate that as , the local heat-flux-maximizing aspect ratio
, the Nusselt number , and the Reynolds number ,
with all prefactors depending on . Moreover, the interior flow of the local
-maximizing solution can be well described by an analytical heat-exchanger
solution, and the connection to the high-wavenumber asymptotic solution given
by Blennerhassett & Bassom is discussed. With a fixed aspect ratio
at , however, our computations show that as
increases, the steady rolls converge to the semi-analytical asymptotic
solutions constructed by Chini & Cox, with scalings and
. Finally, a phase diagram is delineated to gain a
panorama of steady solutions in the high-Rayleigh-number-wavenumber plane