# Prediction of bed level variations in nonuniform sediment bed channel

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## Abstract

In the present study, a numerical model, based on one-dimensional de Saint-Venant equations along with sediment continuity equation, is developed for prediction of bed levels in non-cohesive sediments in aggrading alluvial channels. One-dimensional, unsteady flow equations and sediment continuity equations are solved using ‘shock-capturing’, second order accurate, explicit MacCormack finite difference scheme while considering upstream and downstream boundary conditions in the channel. Series of experimental investigations have been undertaken for measurements of bed and water levels in an aggrading channel due to overloading of nonuniform sediments, extracted from the bed of Tapi River at Surat City, in a flume installed in Advanced Hydraulics Laboratory of SVNIT, Surat, India. A satisfactory coupling between the water flow and sediment flow has been achieved. The sediment continuity equation is used for the each size class to compute the volume of each size class after each time step at any computational node in the computational grid. The fractional bed and suspended load transport capacities for different size fractions have been computed using fractional transport laws for nonuniform sediments. The active bed layer concept has been implemented in finite difference scheme to consider the interaction and exchange of sediment and water flow near the mixing layer. The performance of developed numerical model has been satisfactorily verified with independent experimental data of nonuniform sediment bed. Also, consideration of sediment nonuniformity in computation of bed level variation has been demonstrated by comparing the results based on sediment transport functions of uniform and nonuniform sediments.

## Keywords

Numerical model aggradation non-cohesive sediment mixture active bed layer fractional sediment transport## Notation

*B*_{0}width of channel at initial time line, t = 0

*C*_{n}Courant number

*d*diameter of sediment particle

*d*_{a}arithmetic mean diameter of sediment particle (mm)

*d*_{g}geometric mean size of sediment mixture

*d*_{j}particular size fraction of sediment mixture; geometric mean size of sieve sizes

*d*_{ 1 }and*d*_{ 2 }*d*_{15.9}sizes such that 15.9% of particles are finer, by weight, than these sizes

*d*_{84.1}sizes such that 84.1% of particles are finer, by weight, than these sizes

*d*_{90}sediment size such that 90% of the material, by weight, is finer than this size,

*f*_{i}probability of occurrence of size

*d*_{ i }Under consideration**F**flux vector

*g*acceleration due to gravity

*h*depth of flow

*h*_{0}initial flow depth

*L*length of channel

- M
Kramer’s uniformity coefficient

- M
_{1} represents sediment mixture 1 of the present study

- M
_{2} represents sediment mixture 2 of the present study

*n*Manning’s roughness co-efficient or total number of observations

*p*porosity of bed layer

*p*_{j}percentage of sediment size fraction,

*d*_{ j }, in the*ABL*- Δ
*p*_{j} change in percentage weight corresponding to size

*d*_{ j }*P*_{1}percentage finer by weight for sieve size

*d*_{ 1 }*q*water discharge per unit width

*q*_{0}initial uniform flow discharge

*q*_{T0}rate of equilibrium sediment discharge per unit width of channel

- Δ
*q*_{T} increase in sediment discharge due to overloading

*q*_{B}rate of bed load transported per unit width, by weight, (N/s/m)

*q*_{T}rate of total sediment transport by weight per unit width of channel (N/s/m), (

*q*_{ T }=*q*_{ B }+*q*_{ S })*q*_{Tin}sediment overloading rate at upstream fictitious node (

*q*_{T0}+ Δ*q*_{ T })*q*_{Bj}fractional bed load transport rates of

*j*^{th}size fraction*q*_{sj}fractional suspended load transport rates of

*j*^{th}size fraction- Q
flow discharge (m

^{3}/s)*r*total number of size fractions

*S*_{0}longitudinal slope of the channel at t = 0

**S**source term vector

*S*_{f}friction slope

*t*time scale of computational grid

- t
_{last} last computational step

- Δ
*t* temporal step in y direction

*T*_{b}thickness of active bed layer

- U
average flow velocity

**U**dependent variable vector

*U*variable representing the unknown (h, q and z) for finite difference discretization in present

*V*_{i,j}^{k}total volume of any particle size,

*d*_{ j }, present in the active layer per unit length of the channel, at*i*^{th}space node and*k*^{th}time line*x*distance along the channel

- Δ
*x* space interval in computational grid (m)/distance increment

*z*_{0}initial bed level

*z*bed elevation

- Z
_{0} hydraulic roughness parameter

*z*_{i+1}^{k+1}bed level at computational node i + 1 for unknown time level

*k*+ 1 for each time step

## Greek symbol

*σ*_{g}geometric standard deviations of bed material

*τ*_{0}mean shear stress (N/m

^{2})*τ*_{0c}critical shear stress (N/m

^{2})- є
residual error

## Superscript and subscript

*i*denotes the space node

*j*denotes size fractions in sediment mixture

*k*denotes the time node

*o*stands for initial values

- ∗
superscript values of the variables after the predictor steps at the unknown time k + 1 level

- ∗∗
superscript values of the variables after the corrector steps at the unknown time k + 1 level

## Abbreviations

- ABL
active bed layer

- BL
bed level

- FCM
fully coupled model

- MASE
mean absolute standard error

- RMSE
root mean square error

- SD
standard deviation

- WL
water level

## Notes

### Acknowledgements

The authors would like to thank the Department of Science and Technology (DST), New Delhi, Government of India, for the financial support for installing the flume facilities through the Research Project “Erosion of nonuniform unimodal and bimodal sediments” with project Grant No. SR/S3/MERC/015/2009.

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