具有有效界面条件的均质化模型用于斯托克斯流通过多孔膜的数值研究

中文摘要 AI 辅助

我们数值研究了通过薄多孔层的流体流动，该多孔层被建模为分隔两个体域的低维界面。在该界面上，我们施加了有效的透射条件，这些条件涉及先前工作中通过双尺度方法严格推导的均质化系数。这些系数由编码多孔层微观特性的单元问题决定。第一步，我们定性分析了有效系数如何依赖于孔隙几何形状。基于这些计算，我们随后对包含有效界面条件的宏观模型进行了数值模拟。尽管用有效界面替代薄多孔层大大降低了计算成本，但由此产生的非标准透射条件给数值实现带来了额外挑战。我们采用了一种Taylor-Hood型混合有限元离散化，在界面上具有不连续的压力和切向速度，并证明了所得格式的稳定性和误差估计。最后，对于仍可计算解析的微观几何，我们将宏观模型的结果与微观模型的直接模拟结果进行了比较。

英文摘要

We investigate numerically fluid flow through a thin porous layer, which is modelled as a lower-dimensional interface separating two bulk domains. At this interface, we impose effective transmission conditions involving homogenized coefficients rigorously derived via two-scale methods in previous works. These coefficients are determined by cell problems encoding the microscale properties of the porous layer. In a first step, we analyze qualitatively how the effective coefficients depend on the pore geometry. Based on these computations, we then present numerical simulations of the macroscopic models incorporating the effective interface conditions. Although replacing the thin porous layer by an effective interface substantially reduces the computational cost, the resulting nonstandard transmission conditions pose additional challenges for the numerical implementation. We employ a Taylor--Hood-type mixed finite element discretization with discontinuous pressure and tangential velocity across the interface and prove stability as well as error estimates for the resulting scheme. Finally, for microscale geometries that are still computationally resolvable, we compare the results of the macroscopic models with those obtained from direct simulations of the microscopic models.

发表机构

• Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）
• TU Bergakademie Freiberg（弗莱贝格工业大学）
• Universität Augsburg（奥格斯堡大学）

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