Computation of three-dimensional Brinkman flows using regularized methods

The Brinkman equations of fluid motion are a model of flows in a porous medium. We develop the exact solution of the Brinkman equations for three-dimensional incompressible flow driven by regularized forces. Two different approaches to the regularization are discussed and compared on test problems. The regularized Brinkman model is also applied to the unsteady Stokes equation for oscillatory flows since the latter leads to the Brinkman equations with complex permeability parameter. We provide validation studies of the method based on the flow and drag of a solid sphere translating in a Brinkman medium and the flow inside a cylindrical channel of circular cross-section. We present a numerical example of a swimming organism in a Brinkman flow which shows that the maximum swimming speed is obtained with a small but non-zero value of the porosity. We also demonstrate that unsteady Stokes flows with oscillatory forcing fall within the same framework and are computed with the same method by applying it to the motion of the oscillating feeding appendage of a copepod.