Convection due to an unstable density difference across a permeable membrane

Abstract

We study natural convection driven by unstable concentration differences of sodium chloride (NaCl) across a horizontal permeable membrane at Rayleigh numbers (Ra) of 10¹⁰ to 10¹¹ and Schmidt number (Sc)=600. A layer of brine lies over a layer of distilled water, separated by the membrane, in square-cross-section tanks. The membrane is permeable enough to allow a small flow across it at higher driving potentials. Based on the predominant mode of transport across the membrane, three regimes of convection, namely an advection regime, a diffusion regime and a combined regime, are identified. The near-membrane flow in all the regimes consists of sheet plumes formed from the unstable layers of fluid near the membrane. In the advection regime observed at higher concentration differences (Δ C) across the membrane, there is a slow overturning through-flow across the membrane; the transport across the membrane occurs mostly by advection. This phenomenology explains the observed Nu_b~Ra²/Sc scaling of the Nusselt number. The planforms of sheet plumes near the membrane show a dendritic structure due to the combined influence of the mean shear due to the large-scale flow and the entrainment flow of the adjacent plumes. The near-membrane dynamics show initiation, elongation and merger of plumes; a movie is available with the online version of the paper. Increase in Ra results in a larger number of closely and regularly spaced sheet plumes. The mean plume spacing in the advection regime λ̅_b , is larger than the mean plume spacing in Rayleigh-Bénard convection (λ̅), and shows a different Ra-dependence. The plume spacings in the advection regime (λ_b) show a common log-normal probability density function at all Ra. We propose a phenomenology which predicts λ̅_b ~√ Z_wZ_vi, where Z_w and Z_vi are, respectively, the near-wall length scales in Rayleigh-Bénard convection (RBC) and due to the advection velocity. In the combined regime, which occurs at intermediate values of Δ C, the flux scales as (Δ C/2)^4/3. At lower driving potentials, in the diffusion regime, the flux scaling is similar to that in turbulent RBC.