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.

翻译：电动电化学和电动电动装置的各种电化学和电动装置,其分析、设计和优化都是通过电化学装置(PNP)方程式和Navier-Stokes(NS)方程式来体现的。为了准确捕捉离子浓度和当前通量的时空变异,直接数字模拟(DNS)仍然具有挑战性,因为(a) 电极双层(EDL)的小型关键层面;(b) 硬联动、大反动效应和靠近边界的斜梯度;以及(c) 电化学装置显示的复杂的长期空间变异性。在目前的研究中,我们通过直接数字模拟框架来应对这些挑战,其中包括:(a) 变异多级(VMS)处理,(b) 与半隐含(NS)和隐含(PNP)时间调调)的时间框架相结合;以及(c) 基于电离子结构的二次调整模型改进。VMS的配制提供了数字关键时间级稳定度,用于在电极-NF-S-S-imal-imal-imal Stal-al IMLLisal IMLisal 的精确变变变法中,经常观察到的精确变现变现变法。