Finite element modeling of charged species transport has enabled the
analysis, design, and optimization of a diverse array of electrochemical and
electrokinetic devices. These systems are represented by the
Poisson-Nernst-Planck (PNP) equations coupled with the Navier-Stokes (NS)
equation. Direct numerical simulation (DNS) to accurately capture the
spatio-temporal variation of ion concentration and current flux remains
challenging due to the (a) small critical dimension of the electric double
layer (EDL), (b) stiff coupling, large advective effects, and steep gradients
close to boundaries, and (c) complex geometries exhibited by electrochemical
devices.
In the current study, we address these challenges by presenting a direct
numerical simulation framework that incorporates: (a) a variational multiscale
(VMS) treatment, (b) a block-iterative strategy in conjunction with
semi-implicit (for NS) and implicit (for PNP) time integrators, and (c) octree
based adaptive mesh refinement. The VMS formulation provides numerical
stabilization critical for capturing the electro-convective instabilities often
observed in engineered devices. The block-iterative strategy decouples the
difficulty of non-linear coupling between the NS and PNP equations and allows
using tailored numerical schemes separately for NS and PNP equations. The
carefully designed second-order, hybrid implicit methods circumvent the harsh
timestep requirements of explicit time steppers, thus enabling simulations over
longer time horizons. Finally, the octree-based meshing allows efficient and
targeted spatial resolution of the EDL. These features are incorporated into a
massively parallel computational framework, enabling the simulation of
realistic engineering electrochemical devices. The numerical framework is
illustrated using several challenging canonical examples
