Quantifying the Extent of the Hyporheic Zone Using Pore-Resolved DNS of Turbulent Flow over a Random Sphere Packing

The vertical extent of exchange between surface flow and sediment pore water is a key control on ecological functioning in fluvial systems. Across the diverse disciplines engaged in hyporheic-zone research, however, this exchange depth is characterized using different criteria and definitions. From a hydrodynamic and transport perspective, this raises the question of how far interface-driven scalar transport mechanisms penetrate into a porous sediment bed beneath turbulent open-channel flow.

Using pore-resolved Direct Numerical Simulation (DNS), we investigate scalar transport near the sediment--water interface within the framework of effective diffusivity. The porous medium is represented by a random sphere packing overlain by a turbulent open-channel flow characterized by a friction Reynolds number of Re_τ = 180 and a permeability Reynolds number of Re_K = 1.8. Scalar transport is modeled by solving the advection-diffusion equation for a passive scalar at Schmidt number Sc = 1, subject to a prescribed vertical concentration gradient driving bed-normal transport.

To obtain statistically representative descriptions of the strongly three-dimensional flow and transport fields, double averaging, in time and over horizontal planes, is employed. Within this framework, the effective diffusivity model relates the plane-averaged scalar concentration to the vertical scalar flux, enabling a quantitative decomposition into turbulent transport, dispersive transport, and molecular diffusion. Revisiting the theory of horizontal averaging, we discuss the implications of different formulations of effective diffusivity and show that seemingly minor differences become significant in regions of rapidly varying porosity, such as the sediment--water interface.

In a systematic pre-study, we assess the influence of grid resolution, sampling duration, and sediment-bed depth on the resulting transport statistics. Based on this analysis, simulations are conducted at a constant flow-depth-to-sphere-diameter ratio of h_f / D = 3, combined with three sediment-bed depths corresponding to h_b / D ∈ [2, 5, 9]. To isolate interface-induced transport processes, simulations with overlying turbulent flow are compared to reference cases of scalar transport in the porous medium without free flow.

This comparison enables a clear distinction between transport mechanisms intrinsic to the porous medium and those induced by the sediment--water interface. The effective diffusivity associated with interface-driven transport decays exponentially with increasing depth below the sediment surface. Turbulent scalar transport is confined to the uppermost sediment layer, penetrating only to depths of approximately z / D ≅ 1 - 2. In contrast, dispersive transport induced by pressure fluctuations at the sediment crest dominates scalar exchange within the sediment bed. Beyond z / D ≅ 5, dispersive transport becomes negligible and scalar transport is governed predominantly by molecular diffusion, indicating the onset of a Darcy-type transport regime. Within the effective diffusivity framework, cases with h_b / D = 5 and 9 show indistinguishable behavior, whereas the shallow bed case h_b / D = 2 exhibits a pronounced attenuation of dispersive transport.

These results provide a quantitative, transport-based definition of the effective depth of interface-driven hyporheic exchange. The exponential decay of effective diffusivity, isolated from background porous-medium transport, offers a promising basis for improving reduced-order models of scalar transport in the hyporheic zone.