Computational 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 diffuse charge layer (DCL), (b) stiff coupling due to fast charge relaxation times, 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 flows often observed in engineered devices. The block-iterative strategy decouples the difficulty of non-linear coupling between the NS and PNP equations and allows the use of 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 DCL. 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：带电物种输运的计算建模已能够对多种电化学和电动装置进行分析、设计与优化。这些系统由泊松-能斯特-普朗克（PNP）方程与纳维-斯托克斯（NS）方程耦合描述。由于（a）扩散电荷层（DCL）的临界尺寸极小，（b）快速电荷弛豫时间、强对流效应以及边界附近的陡峭梯度导致的刚性耦合，以及（c）电化学装置呈现的复杂几何结构，准确捕捉离子浓度和电流通量时空变化的直接数值模拟（DNS）仍具挑战性。在本研究中，我们通过提出一个直接数值模拟框架来应对这些挑战，该框架整合了（a）变分多尺度（VMS）处理，（b）结合半隐式（用于NS方程）和隐式（用于PNP方程）时间积分器的块迭代策略，以及（c）基于八叉树的自适应网格细化。VMS方法提供了数值稳定化，对于捕捉工程装置中常见电对流流动至关重要。块迭代策略解耦了NS与PNP方程非线性耦合的困难，并允许对NS和PNP方程分别使用定制数值格式。精心设计的二阶混合隐式方法规避了显式时间步进器的苛刻时间步长要求，从而能够在更长时间跨度内进行模拟。最后，八叉树网格划分实现了对DCL的高效且有针对性的空间分辨率。这些特性被整合到一个大规模并行计算框架中，从而能够模拟实际工程电化学装置。我们通过几个具有挑战性的典型算例展示了该数值框架。