Addressing bedload flux variability due to grain shape effects and experimental channel geometry

TL;DR

This study tackles the persistent variability in bedload flux measurements by establishing a universal channel-geometry correction that robustly determines the transport-driving bed shear stress $\tau_b$ across narrow, shallow, and wide channels. It combines a physically grounded bed-surface definition with a Kolmogorov-based sidewall correction, enabling consistent collapse of spherical-grain data across experiments and grain-resolved CFD-DEM simulations. The authors then extend a physically based bedload flux model to account for grain-shape variability and lift forces, validating it against an extensive data compilation that spans channel widths, slopes, depths, and grain shapes; the model predicts most data within a factor of $1.3$. Overall, the work reduces bedload-transport variability arising from geometry and shape effects, improving predictive capability for sediment-transport processes in natural and engineered channels.

Abstract

The study-to-study variability of bedload flux measurements in turbulent sediment transport borders an order of magnitude, even for idealized laboratory conditions. This uncertainty stems from physically poorly supported, empirical methods to account for channel geometry effects in the determination of the transport-driving bed shear stress and from study-to-study grain shape variations. Here, we derive a universal method of bed shear stress determination. It consists of a physically-based definition of the bed surface and a channel sidewall correction that does not rely on empirical elements, except for well-established scaling coefficients associated with Kolmogórov's theory of turbulence. Application of this method to bedload transport of spherical grains -- to rule out grain shape effects -- collapses data from existing laboratory measurements and grain-resolved CFD-DEM simulations for various channel geometries onto a single curve. In contrast, classical sidewall corrections, as well as an alternative bed surface definition, are unable to universally capture these data, especially those from shallow or very narrow channel flows. We then apply our method to an extended grain-shape-controlled data compilation, complemented by literature data for non-spherical grains and from grain-unresolved CFD-DEM simulations. This compilation covers a very diverse range of transport conditions, ranging from very narrow to infinitely wide channels, from shallow to deep channel flows, from mild to steep bed slopes, and from weak to intense transport. We generalize an existing physical bedload flux model to account for grain shape variability and show that it explains almost all the compiled data within a factor of only $1.3$.

Figures (12)

• Figure 1: Bedload flux measurements for spherical grains by NiCapart18 and Dealetal23a. The Shields number $\Theta$ and nondimensionalized bedload flux $Q_\ast$ were defined and determined as described in these studies. For $\Theta\approx0.2$, the values of $Q_\ast$ disagree from each other by a factor of about $6$.
• Figure 2: Determination of bed surface elevation for experimental data set NC18EXP NiCapart18. The vertical profiles of $\dot\gamma\Sigma\phi$ exhibit several local maximums due to layering. We choose the top-most local maximum as the bed surface elevation because the vertical profile of the fluid shear stress $\tau^f_{zx}(z)$ exhibits a focal point there, with $\tau^f_{zx}(0)\approx\tau_t$ (§\ref{['FurtherTests']}), consistent with the theoretical expectation PahtzDuran18b.
• Figure 3: Channel geometry corrections. Nondimensionalized bedload flux $Q_\ast$ versus Shields numbers for spherical grains: (A) uncorrected for sidewall effects ($\Theta_0$), (B) hydraulic-radius correction Guo15 ($\Theta_R$), (C) Einstein-Johnson correction Einstein42Johnson42VanoniBrooks57Guo17 ($\Theta_E$), (D) our universal correction ($\Theta$). Symbols correspond to data from experiments and grain-resolved simulations (Table \ref{['Data']}). An error bar indicates the standard error and/or uncertainty range. In the absence of error bars, uncertainties are smaller than the symbol size. Except for KU17DNS ($s^{1/2}Ga\simeq44.86$, smooth), all data are in the rough regime ($s^{1/2}Ga>70$). The values of $h$ used to determine $\tau_b$ in Z22LESw and NC18EXP are based on a granular-physics-based bed surface definition PahtzDuran18b, equation (\ref{['BedSurface']}), and in NC18EXP$\ast$ on the original definition reported by NiCapart18. For all other data sets, reported values of $h$ are used as it has a negligible effect on the Shields numbers. Z22LESn$\ast$ refers to unmodified data, whereas for Z22LESn in (D), the sidewall correction, and thus $\Theta$, has been modified to account for the fact that the grid resolution of the grain-resolved LES simulations behind these data was limited to $15$ times the Kolmogórov length scale (see text).
• Figure 4: Channel geometry corrections. (A) $\Theta$ versus $2.41\Theta_R$ (using $d=c_g$ for non-spherical-grain data, as explained in §\ref{['BedloadFluxModel']}), based on $\tau_b=2.41\tau_R$, the empirical correction in Dealetal23a. (B) Log-profile shear velocity $u_\tau\simeq\sqrt{\tau_b/\rho_f}$ predicted by our sidewall correction versus measured one for experimental data from particle-free flows in open channels consisting of rough beds and smooth sidewalls Song94SongChiew01Aueletal14 (data as summarized in the tables of Guo15). For nonuniform flows, the bed slope $\tan\alpha$ is corrected using equation (1b) of Guo15. The roughness sizes are $r=c_g/2$ for the experiments of Song94 (water-worked bed), $r=d_o$ for those of SongChiew01 (sand grains glued on aluminum plate), and $r=k$ for those of Aueletal14 (with $k$ the measured absolute roughness).
• Figure 5: Independent evidence supporting our bed surface elevation definition. Vertical profiles (solid lines) of the nondimensionalized fluid shear stress $\Theta^f_{zx}\equiv\tau^f_{zx}/(\rho_p\tilde{g}_zd)$ for the data set NC18EXP NiCapart18. The data exhibit a focal point at the bed surface elevation $z=0$ defined by equation (\ref{['BedSurface']}): $\Theta^f_{zx}(0)\approx0.04$, a property required for Bagnoldian-type bedload models PahtzDuran18b. The fluid shear stress $\tau^f_{zx}$ is calculated from local quantities at the channel center as described in Chauchat18, assuming that the mixture viscosity obeys the closure measured by Boyeretal11 for viscous suspensions. The dashed lines correspond to the bed surface elevations determined by NiCapart18.