Computational homogenization of unsteady flows with obstacles

Abstract

AbstractModeling flows in domains containing obstacles becomes challenging when the obstacles are very small compared to the domain. This problem is often known as modeling flow through permeable or porous media. Darcy's law is well known to provide a robust solution to this problem for viscous flows, given that the permeability of the medium can be characterized. Solutions for inertial flows have not been proposed yet. In this article, a principle multiscale virtual power is formulated for the computational homogenization of unsteady incompressible flows in domains containing small obstacles. In this theory, the coarse scale of the domain is separated from the fine scale of the obstacles. The finite element (FE) implementation of this theory is developed in the form of a parallel (FE) algorithm with two‐way coupling between the two scales. The incompressibility constraint is handled with special care by introducing an independent pressure variable at both scales and relying on the Taylor–Hood P2/P1 pair. Simulations are conducted to analyze the accuracy and efficiency of the proposed multiscale approach. The results show that the method is robust and interesting from both the points of view of computational cost and accuracy when the ratio between the domain size and the obstacle size is larger than 100.