Rayleigh-Bénard percolation transition of thermal convection in porous media: Computational fluid dynamics, NMR velocity mapping, NMR temperature mapping

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Abstract
Stationary thermal convection and heat conduction were studied in random-site percolation clusters in a Rayleigh-Bénard configuration experimentally by NMR microscopy techniques and numerically with the aid of a finite volume method. The porosity of the percolation clusters, and the temperature difference applied to the convection cell were varied. Two-dimensional percolation networks were generated with the aid of a random-number algorithm. The resulting clusters were used as templates for the fabrication of model objects. The convective velocity distribution of silicon oil or ethylene glycol filled into the pore space was mapped and evaluated in the form of histograms. The flow patterns visualized in the simulated and the measured velocity maps show good coincidence. In the histograms, two velocity regimes can be distinguished and attributed to local convection rolls responsible for the low-velocity part, and cluster-spanning flow loops characterized by a high-velocity cut-off, respectively. The maximum velocity as a function of the porosity and the overall temperature difference is shown to be indicative for the hydro-thermodynamic Rayleigh-Bénard instability and the geometrical percolation threshold. The coincidence of the Rayleigh-Bénard instability (modified for porous media) and the percolation transition (modified for closed loops) gives rise to a new critical phenomenon termed the Rayleigh-Bénard percolation transition. It occurs at a certain combination of the porosity and the overall temperature difference in the cell. Temperature maps were recorded with the aid of a relaxation-based NMR technique. The consequence of different thermal conductivities in the matrix and in the fluid is that horizontal-temperature gradients arise even in the absence of flow. This leads to a superposition of uncritical convective flow driven by the horizontal-temperature gradients whenever closed-loop pathways are possible and the critical Rayleigh-Bénard convection based on vertical-temperature gradients.