Application of an Immersed Boundary Method to Flow in Cerebral Aneurysms and Porous Media

We present the development and application of an immersed boundary (IB) method for the simulation of incompressible flow inside and around complex geometrical shapes and cavities. The IB method is based on a volume-penalization method that is applied throughout the domain, rendering the velocity in stationary solid parts negligibly small, while the flow in the open parts of the domain is governed by the Navier-Stokes equations. The flow solver is based on a skew-symmetric finite-volume discretization in combination with explicit time-stepping for the convective and viscous fluxes, and implicit time-stepping for the IB forcing term. The complex geometry is characterized in terms of a so-called ‘masking function’ which equals unity in the solid parts and zero in the open parts of the domain. The focus is on the accuracy with which gradients of the solution close to solid walls can be approximated using the IB methodology. We investigate this for flow through a model of an aneurysm as may develop in the circle of Willis in a human brain, and to flow in a structured porous medium composed of a regular spatial arrangement of square rods. The shear stress acting on the vessel walls in case of flow through an aneurysm and the permeability of the porous material were analyzed. The computational method converges as a first order method for Poiseuille flow, with a considerable influence derived from the precise definition of the masking function near solid-fluid interfaces. We identify the best masking function strategy and show that for plane Poiseuille flow even second order convergence may be obtained. Qualitatively reliable results are obtained already at modest resolutions of 8-16 grid cells across a characteristic opening in the flow domain, e.g., the vessel diameter or the size of the gap between individual square rods.