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

Abstract

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 × 10 4 . The dependence of the bounds on L ( Ra ) is explored and a “minimal flow unit” is identified. • DNS is used to quantify the aspect-ratio ( L ) dependency of Nu ( Ra ) in porous medium convection. • A “minimal flow unit” for porous medium convection is identified. • Rigorous upper bounds on Nu ( Ra ) are computed as a function of L for Ra ⩽ 2.65 × 10 4 . • A numerical scheme is developed for solving upper bound variational problems without continuation.