Fluids moving through porous materials are often assumed to follow Darcy's law when the flow is slow. At higher speeds, inertia becomes important, and the flow becomes nonlinear, but how this shift actually occurs inside the pores has remained unclear. The commonly used criteria for identifying flow nonlinearity, such as the recirculation zone or the Reynolds number (Re), show significant variability across different studies, indicating a lack of consistency and robustness in current approaches. Here, two-dimensional pore-scale simulations directly solving the Navier–Stokes equations are used to examine the Darcy–non-Darcy transition at low Reynolds numbers. The results suggest that nonlinear behavior may begin earlier than previously expected, possibly even before visible eddies or recirculation zones appear. To provide a more sensitive and physically grounded measure, we introduce a dimensionless circulation number IΩ, defined as the global circulation normalized by a characteristic convective scale. Unlike recirculation volume, which is localized and threshold-dependent, IΩ offers a domain-integrated quantification of rotational intensity. This formulation enables detection of nonlinear inertial effects even when macroscopic eddies are not yet fully developed, thereby providing a robust diagnostic of the viscous–inertial transition. The scaling relationship between IΩ and the Reynolds number reveals three regimes: a viscosity-dominated regime, a viscous–inertial transition regime, and an inertia-dominated regime. Contrary to the notion of a single critical point separating Darcy–non-Darcy flow, we identify a continuous viscous–inertial transition in which competition between vorticity amplification and viscous diffusion produces non-periodic flow fluctuations. The IΩ–Re relation provides a sensitive pore-scale diagnostic for flow-regime transitions.
Cheng et al. (2026) studied this question.