Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porous medium convection

Direct numerical simulation (DNS) has shown that Rayleigh-Bénard convection in a fluid-saturated porous medium self-organizes into narrowly spaced plumes at (ostensibly) asymptotically high values of the Rayleigh number Ra. In this Letter a combination of DNS and upper bound theory is used to investigate the dependence of the Nusselt number Nu on the domain aspect ratio L at large Ra. A novel algorithm is introduced to solve the optimization problems arising from the upper bound analysis, allowing for the best available bounds to be extended up to Ra≈2.65×104. The dependence of the bounds on L(Ra) is explored and a “minimal flow unit” is identified.