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Effects exerted by average particle size and non-uniformity on bed surface fractal properties

  • Research Article - Hydrology
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Abstract

Sediment bed surfaces exist widely in natural rivers, and many aspects in river dynamics are closely relevant to bed surface roughness, such as flow structure, river resistance and sediment transport. As two important parameters for quantifying bed surface roughness, how average particle size and non-uniformity affect bed surface structure is unknown. Therefore, nine groups of sediment samples with different average particle sizes or different non-uniformities were firstly prepared by screening dry natural sediments. Then, the prepared sediment samples were used to manually pave nine groups of bed surfaces, and the high-precise bed surface digital elevations were obtained by a handheld 3D laser scanner. Finally, the effects exerted by the average particle size and non-uniformity on the bed surface fractal properties were discussed. The results showed that there is only a scale-free range in a profile or a two-dimensional specific direction of a bed surface with normal-distributed particle gradation. The averaged scale-free upper limit in the two-dimensional specific directions and that related to many profiles are less affected by the non-uniformity, but more affected by the average particle size. For the bed surfaces with the same non-uniformity, when the average particle size is smaller than 15 mm, the larger the average particle size is, the smaller the fractal dimension is, but the larger the scale coefficient is; when the average particle size is larger than 15 mm, the larger the average particle size is, the larger the fractal dimension and the scale coefficient are, while for the bed surfaces with the same average particle size, the non-uniformity has no significant effects on the fractal dimension and the scale coefficient. The averaged scale coefficient in the two-dimensional specific directions of an isotropic bed surface and that related to many profiles are approximately equal, but the averaged fractal dimension in the two-dimensional specific directions is obviously larger than that plus 1 related to many profiles.

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Data availability

All data used during the study are available from the corresponding author by request.

Abbreviations

A :

Average standard deviation (in formulas (10) and (12))

C p :

Profile scale coefficient (in formulas (5) and (7))

\(\overline{{C_{p} }}\) :

Averaged scale coefficient related to many profiles (in formula (9), in Figs. 10 and 11)

C s :

Bed surface scale coefficient in a two-dimensional specific direction (in formulas (2) and (4))

\(\overline{{C_{S} }}\) :

Averaged scale coefficient in many two-dimensional specific directions (in formula (9), in Figs. 9 and 10)

d :

Particle size (in Table 1, in Fig. 3)

D p :

Profile fractal dimension (in formulas (5) and (6))

\(\overline{{D_{p} }}\) :

Averaged fractal dimension related to many profiles (in formula (8), in Figs. 10 and 11)

D s :

Bed surface fractal dimension in a two-dimensional specific direction (in formulas (2) and (3))

\(\overline{{D_{S} }}\) :

Averaged fractal dimension in many two-dimensional specific directions (in formula (8), in Figs. 9 and 10)

F c :

Fractal discriminant number (in formula (12))

R 2 :

Correlation coefficient (in formulas (11) and (12))

x, y, z :

Transverse coordinate, longitudinal coordinate, bed surface elevation

\(\mu_{d}\) :

Mean of particle sizes (average particle size, in Table 1, in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)

\(\sigma_{d}\) :

Standard deviation of particle sizes (non-uniformity, in Table 1, in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)

\(\gamma (\vec{h})\) :

Semi-variogram (in formulas (1), (2), (5), (10) and (11))

\(\vec{h}\) :

Displacement vector (in formulas (1), (2) and (5))

h x, h y :

Components of a displacement vector in the x-axis and y-axis directions (in formulas (1) and (2), in Figs. 5 and 6)

\(\vec{i},\;\vec{j}\) :

Unit vectors in the x-axis and y-axis directions (in formula (1))

\(|\vec{h}|\) :

Calculation interval (\(= \sqrt {h_{x}^{2} + h_{y}^{2} }\); in formulas (2), (5) and (10) –(12))

\(|\overrightarrow {{h_{cp} }} |\) :

Profile scale-free upper limit (in Figs. 5 and 6)

\(\overline{{|\overrightarrow {{h_{cp} }} |}}\) :

Averaged scale-free upper limit related to many profiles (in Fig. 8)

\(|\overrightarrow {{h_{cs} }} |\) :

Bed surface scale-free upper limit in a two-dimensional specific direction (in Figs. 5 and 6)

\(\overline{{|\overrightarrow {{h_{cs} }} |}}\) :

Averaged scale-free upper limit in many two-dimensional specific directions (in Fig. 7)

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Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 51979181, 51539007 and 51279117).

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Authors and Affiliations

  1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, China

    Yunwen Pan & Junqiang **a

  2. State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, 610065, China

    Yunwen Pan, Tun Cai & Kejun Yang

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  1. Yunwen Pan
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  2. Junqiang **a
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  3. Tun Cai
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Contributions

YP involved in writing the original draft; JX took part in review and revising; TC involved in conducting the experiment; KY involved in funding acquisition and submitting the paper. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Kejun Yang.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Edited by Dr. Michael Nones (CO-EDITOR-IN-CHIEF).

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Pan, Y., **a, J., Cai, T. et al. Effects exerted by average particle size and non-uniformity on bed surface fractal properties. Acta Geophys. 71, 517–529 (2023). https://doi.org/10.1007/s11600-022-00888-3

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