Assessment of the Darcy-Brinkman model for the characterisation of flows in permeable substrates

In this work we aim to characterise the flow within matrices of parallel fibres, motivated by their potential for turbulent drag reduction (Gomez-de-Segura et al., J. Fluid Mech., vol 875, 2019, pp. 124-172). For this we conduct fully resolved simulations of the microscale flow within the substrate, driven by shear and/or pressure, both along and across the fibres, for pore sizes small enough to assume Stokes flow. As expected, the component of the flow driven by pressure is well characterised by a Darcy description, but we also show that the component driven by shear follows a self-similar, exponentially decaying form, and is thus well characterised by a Brinkman description. This is the case for the longitudinal flow for fibres of any size, and for the transverse flow for fibres of small size. Beyond a threshold size, however, although the flow remains exponential in magnitude, it experiences regions of reversed direction, which a Brinkman description would fail to capture. For structured arrangements of fibres, the macroscopic Darcy-Brinkman description breaks down near interfaces with free-flow regions, and a jump in flow variables, particularly in shear, is observed, which can be traced to the stress exerted on the flow by the last row of fibres. For disorganised arrangements, however, no such discrete row of fibres exists, no jump in flow variables is observed, and therefore a Darcy-Brinkman description characterises well both the flow deep within the substrate and at the interface.