Towards unified drag laws for inertial flow through fibrous materials

We give a comprehensive survey of published experimental, numerical and theoretical work on the drag law correlations for fluidized beds and flow through porous media, together with an attempt of systematization. Ranges of validity as well as limitations of commonly used relations (i.e. the Ergun and Forchheimer relations for laminar and inertial flows) are studied for a wide range of porosities. The pressure gradient is linear in superficial velocity, U for low Reynolds numbers, Re, referred to as Darcy’s law. Here, we focus on the non-linear contribution of inertia to the transport of momentum at the pore scale, and explain why there are different non-linear corrections on the market. From our fully resolved finite element (FE) results, for both ordered and random fibre arrays, (i) the weak inertia correction to the linear Darcy relation is third power in U, up to small Re ∼ 1–5. When attempting to fit our data with a particularly simple relation, (ii) a non-integer power law performs astonishingly well up to the moderate Re ∼ 30. However, for randomly distributed arrays, (iii) a quadratic correction performs quite well as used in the Forchheimer (or Ergun) equation, from small to moderate Re. Finally, as main result, the macroscopic properties of random, fibrous porous media are related to their microstructure (arrangement) and porosity. All results (Re < 30) up to astonishingly large porosity, ε ∼ 0.9, scale with Reg, i.e., the gap Reynolds number that is based on the average second nearest neighbour (surface to surface) distances. This universal result is given as analytical closure relation, which can readily be incorporated into existing (non)commercial multi-phase flow codes. In the transition regime, the universal curve actually can be fitted with a non-integer power law (better than ∼1% deviation), but also allows to define a critical Regc ∼ 1, below which the third power correction holds and above which a correction with second power fits the data considerably better.