Evaluation of interfacial fluid dynamical stresses using the immersed boundary method

The goal of this paper is to examine the evaluation of interfacial stresses using a standard, finite difference based, immersed boundary method (IMBM). This calculation is not trivial for two fundamental reasons. First, the immersed boundary is represented by a localized boundary force which is distributed to the underlying fluid grid by a discretized delta function. Second, this discretized delta function is used to impose a spatially averaged no-slip condition at the immersed boundary. These approximations can cause errors in interpolating stresses near the immersed boundary.
To identify suitable methods for evaluating stresses, we investigate three model flow problems at very low Reynolds numbers. We compare the results of the immersed boundary calculations to those achieved by the boundary element method (BEM). The stress on an immersed boundary may be calculated either by direct evaluation of the fluid stress (FS) tensor or, for the stress jump, by direct evaluation of the locally distributed boundary force (wall stress or WS). Our first model problem is Poiseuille channel flow. Using an analytical solution of the immersed boundary formulation in this simple case, we demonstrate that FS calculations should be evaluated at a distance of approximately one grid spacing inward from the immersed boundary. For a curved immersed boundary we present a procedure for selecting representative interfacial fluid stresses using the concepts from the Poiseuille flow test problem. For the final two model problems, steady state flow over a bump in a channel and unsteady peristaltic pumping, we present an 'exclusion filtering' technique for accurately measuring stresses. Using this technique, these studies show that the immersed boundary method can provide reliable approximations to interfacial stresses.