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Application of hausdorff fractal derivative to the determination of the vertical sediment concentration distribution

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International Journal of Sediment Research 2023年第1期38卷 12-23页

作者： Hongguang Sun Shiqian Nie Aaron I.Packman Yong Zhang Dong Chen Chengpeng Lu Chunmiao Zheng State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering Hohai University College of Mechanics and Materials Hohai University College of Mathematics and Systems Science Shandong University of Science and Technology Department of Civil and Environmental Engineering Northwestern University Department of Geological Sciences University of Alabama Institute of Geographic Sciences and Natural Resources Research Chinese Academy of Sciences College of Hydrology and Water Resources Hohai University School of Environmental Science and Engineering Southern University of Science and Technology

The Rouse formula and its variants have been widely used to calculate the steady-state vertical concentration distribution for suspended sediment in steady sediment-laden flows, where the diffusive flux is assumed to be Fickian. Turbulent flow, however, exhibits fractal properties, leading to non-Fickian diffusive flux for sediment particles. To characterize non-Fickian dynamics of suspended sediment, the current study proposes a hausdorff fractal derivative based advection-dispersion equation(HADE)model, where the Fickian diffusive flux in the Rouse model is replaced by a fractal derivative re-scaled using a constant diffusivity. The order of the hausdorff fractal derivative is designed to characterize the influence of the multi-fractal turbulence structure on sediment diffusion. Applications show that the HADE model, with the analytical solution expressed using a stretched exponential function, can accurately describe the observed vertical concentration profiles for suspended sediment with different sizes.This improvement well captures the non-exponential decay of the vertical sediment concentration in turbulent flow. Further analyses of measured sediment concentration profiles reveal that the hausdorff fractal order decreases with the Rouse parameter, which describes the stronger impact of turbulent flow and a more uniform sediment concentration profile for smaller particles. Model comparisons also show that the HADE model provides better performance in describing the sediment concentration profiles than the improved Rouse formula and the standard fractional derivative advection-dispersion equation(FADE), which either under-or over-estimates vertical displacement of sediment particles, likely due to coherent turbulent structures.

关键词： Anomalous diffusion hausdorff fractal derivative Vertical concentration distribution Suspended sediment Metric transform

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