Transport in porous media has many practical applications in science and engineering. This work focuses on the development of numerical methods for analyzing porous media flows and uses two major applications, metal foams and the human lung, to demonstrate the capabilities of the methods. Both of these systems involve complex pore geometries and typically involve porous domains of complex shape. Such geometric complexities make the characterization of the relevant effective properties of the porous medium as well as the solution of the governing equations in conjugate fluid-porous domains challenging. In porous domains, there are typically too many individual pores to consider transport processes directly;​ instead the governing equations are volume-averaged to obtain a new sets of governing equations describing the conservation laws in a bulk sense. There are, however, unknown pore-level terms remaining in the volume-averaged equations that must be characterized using effective properties that account for the effects of processes at the pore level. Once closed, the volume-averaged equations can be solved numerically, however, currently available numerical methods for conjugate domains do not perform well at fluid-porous interfaces when using unstructured grids.

In light of the preceding discussion, the goals of this work are:​ (i) to develop a finite-volume-based numerical method for solving fluid flow and non-equilibrium heat transfer problems in conjugate fluid-porous domains that is compatible with general unstructured grids, (ii) to characterize the relevant flow and thermal properties of an idealized graphite foam, (iii) to determine the permeability of an alveolated duct, which is considered as a representative element of the respiratory region of the human lung, and (iv) to conduct simulations of airflow in the human lung using a novel fluid-porous description of the domain. Results show that the numerical method that has been developed for conjugate fluid-porous systems is able to maintain accuracy on all grid types, flow directions, and flow speeds considered. This work also introduces a comprehensive set of correlations for the effective properties of graphite foam, which will be useful for studying the performance of devices incorporating this material. In order to model air flow in the lung as a porous medium, the permeability of an alveolated duct is obtained using direct pore-level simulations. Finally, simulations of air flow in the lung are presented which use a novel fluid-porous approach wherein the upper airways are considered as a pure fluid region and the smaller airways and alveoli are considered as a porous domain.