Statistical analysis of electroconvection near an ion-selective membrane in the highly chaotic regime

We investigate electroconvection and its impact on ion transport in a model system comprised of an ion-selective membrane, an aqueous electrolyte, and an external electric field applied normal to the membrane. We develop a direct numerical simulation code to solve the governing Poisson-Nernst-Planck and Navier-Stokes equations in three dimensions using a specialized parallel numerical algorithm and sufficient resolution to capture the high frequency and high wavenumber physics. We show a comprehensive statistical analysis of the transport phenomena in the highly chaotic regime. Qualitative and quantitative comparisons of two-dimensional (2D) and 3D simulations include prediction of the mean concentration fields as well as the spectra of concentration, charge density, and velocity signals. Our analyses reveal a significant quantitative difference between 2D and 3D electroconvection. Furthermore, we show that high-intensity yet short-lived current density hot spots appear randomly on the membrane surface, contributing significantly to the mean current density. By examining cross correlations between current density on the membrane and other field quantities we explore the physical mechanisms leading to current hot spots. We also present analysis of transport fluxes in the context of ensemble-averaged equations. Our analysis reveals that in the highly chaotic regime the mixing layer (ML), which spans the majority of the domain extent, is governed by advective fluctuations. Furthermore, we show that in the ML the mean electromigration fluxes cancel out for positive and negative ions, indicating that the mean transport of total salt content within the ML can be represented via the electroneutral approximation. Finally, we present an assessment of the importance of different length scales in enhancing transport by computing the cross covariance of concentration and velocity fluctuations in the wavenumber space. Our analysis indicates that in the majority of the domain the large scales contribute most significantly to transport, while the effects of small scales become more appreciable in regions very near the membrane.