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Quantification of probabilistic concentrations for mixed-size sediment particles in open channel flow

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Abstract

The Rouse equation is a well-known deterministic model for suspended concentrations. However, the transport of sediment particles is influenced by several random variables, such as non-uniform sediment size and turbulence structure. Experiments have demonstrated that the stochastic characteristics of turbulence structure, such as ejections and sweeps, can cause fluctuations in sediment concentrations. A new method is proposed to quantify the probabilistic sediment concentrations. In this study, the multiple-state discrete-time Markov chain and stochastic particle tracking model were used to simulate sediment transport with spatially and temporally varying probabilistic concentrations under the stochastic turbulence structure. Point estimate methods were adopted to estimate the variability of non-uniform sediment sizes. The proposed model was implemented for three cases. In the first case, the proposed model was validated against the experimental data. In the second case, spatial and temporal concentrations at high and low Rouse numbers with mean and non-uniform sediment sizes were compared. The result demonstrates that in the prediction with mean sediment sizes, the sediment concentration is overestimated near the bed, and the advection of the sediment concentration in the x-direction is underestimated. In the last case, higher-order statistical moments of the fluctuating concentrations were estimated through simulations using the proposed model. Simulation results conducted using the proposed method were compared with experimental data. The results revealed that the prediction results based on the proposed model are in good agreement with experimental data.

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Abbreviations

p :

The Matrix of transition probability

\( p_{ij} \) :

Transition probability

\( {\text{N}}^{t} \) :

The state at t

\( \cup \) :

Eigenvector matrix

\( \Lambda ^{t} \) :

Eigenvalue diagonal matrix

k:

segment

l :

Layer

\( \uppi \) :

Stationary probabilities

r:

Coefficient of determination

\( {\text{M}}_{1} \) :

Minimum acceptable time step

v :

Kinematic viscosity

\( {\text{D}}_{x} \) :

The x quantile value of sediment distribution

\( \upxi_{x} \) :

The x number of weighting function

\( \overline{G\left( x \right)} \) :

The average output of the model

g:

Gravitational acceleration

\( \rho \) :

Density of fluid

\( N_{k,l}^{t} \) :

The number of the particle in kth segment, lth layer at time t

\( u_{k,l}^{t} \) :

The input number of the particle in kth segment, lth layer at time t

\( \lambda_{k,l}^{t} \) :

The output number of the particle in kth segment, lth layer at time t

\( u_{*} \) :

Shear velocity

\( w_{s} \) :

Settling velocity

\( \upvarepsilon_{z} \) :

Diffusion coefficient

H:

Water Depth

\( \upalpha \) :

Coefficient in the Fractional diffusion equation

\( \bar{u} \) :

Mean velocity

\( {\text{M}}_{2} \) :

Maximum acceptable time step

B:

The constant value in Logarithmic velocity profile

\( w_{x} \) :

The x number of representative weighting function

\( c_{i} \) :

ith moments of the random variable

\( d_{*} \) :

Dimensionless grain size

\( \rho_{s} \) :

Density of the particle

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Acknowledgements

This study was supported by the Taiwan Ministry of Science and Technology (MOST) under Grant No. 104-2628-E-002-011-MY3 and 108-2221-E-002-011-MY3, as well as by the Taiwan Ministry of Education under NTU Fund No. 109L7843.

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  1. Department of Civil Engineering, National Taiwan University, Taipei, Taiwan

    Chi-Hsiang Huang, Christina W. Tsai & Seyyed Mahmoud Mousavi

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  1. Chi-Hsiang Huang
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  2. Christina W. Tsai
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  3. Seyyed Mahmoud Mousavi
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Correspondence to Christina W. Tsai.

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Huang, CH., Tsai, C.W. & Mousavi, S.M. Quantification of probabilistic concentrations for mixed-size sediment particles in open channel flow. Stoch Environ Res Risk Assess 35, 419–435 (2021). https://doi.org/10.1007/s00477-020-01886-x

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