Measurement of an eddy diffusivity for chaotic electroconvection using combined computational and experimental techniques

The coupling between ion transport, fluid flow, and electrostatics may give rise to electroconvection, a physical phenomenon in which the buildup of charge near selective surfaces leads to hydrodynamic instability, eventually transitioning to an unsteady and chaotic flow. Though electroconvection contains a wide range of spatiotemporal scales, averaging of the ion concentration, velocity, and electric potential fields may produce a lower-dimensional and smoother representation while still capturing the essential performance metrics: ion current density and mean applied voltage, for example. The Poisson-Nernst-Planck-Stokes equations are known to capture the chaotic dynamics of electroconvection accurately. However, there is as of yet no way to directly compute the mean fields, since the application of the well-known Reynolds-averaging procedure leads to a closure problem. In this work, we combine the macroscopic forcing method, a numerical technique for measurement of closure operators in Reynolds-averaged equations, with high-fidelity experimental data to close the equations for the mean fields in chaotic electroconvection. We show that the unclosed fluxes in the Reynolds-averaged equations may be represented to the leading order as a gradient-diffusion term, with a spatially varying eddy diffusivity that we directly measure from experimental velocity fields. As a result, we are able to directly solve for the mean ion concentration and electric potential fields by supplementing the 1D Poisson-Nernst-Planck equations with an eddy diffusivity that captures the averaged effects of mixing due to chaotic, 3D electroconvection. The resulting current-voltage curve exhibits strong agreement with experiments. Our method allows for the study of electroconvection with orders -of -magnitude cost savings by obviating the need for expensive direct numerical simulations. Additionally, our measured eddy diffusivity profiles provide a benchmark for the development of stand-alone reduced -order models of electroconvection.