Depth-Averaged Modelling of Erosion and Sediment Transport in Free-Surface Flows

A fast finite volume solver for multi-layered shallow
water flows with mass exchange and an erodible bed is developed.
This enables the user to solve a number of complex sediment-based
problems including (but not limited to), dam-break over an erodible
bed, recirculation currents and bed evolution as well as levy and
dyke failure. This research develops methodologies crucial to the
under-standing of multi-sediment fluvial mechanics and waterway
design. In this model mass exchange between the layers is allowed
and, in contrast to previous models, sediment and fluid are able
to transfer between layers. In the current study we use a two-step
finite volume method to avoid the solution of the Riemann problem.
Entrainment and deposition rates are calculated for the first time in
a model of this nature. In the first step the governing equations are
rewritten in a non-conservative form and the intermediate solutions
are calculated using the method of characteristics. In the second stage,
the numerical fluxes are reconstructed in conservative form and are
used to calculate a solution that satisfies the conservation property.
This method is found to be considerably faster than other comparative
finite volume methods, it also exhibits good shock capturing. For most
entrainment and deposition equations a bed level concentration factor
is used. This leads to inaccuracies in both near bed level concentration
and total scour. To account for diffusion, as no vertical velocities
are calculated, a capacity limited diffusion coefficient is used. The
additional advantage of this multilayer approach is that there is a
variation (from single layer models) in bottom layer fluid velocity:
this dramatically reduces erosion, which is often overestimated in
simulations of this nature using single layer flows. The model is
used to simulate a standard dam break. In the dam break simulation,
as expected, the number of fluid layers utilised creates variation in
the resultant bed profile, with more layers offering a higher deviation
in fluid velocity . These results showed a marked variation in erosion
profiles from standard models. The overall the model provides new
insight into the problems presented at minimal computational cost.

Authors:

• Thomas Rowan
• Mohammed Seaid

Keywords:

• Erosion
• finite volume method
• sediment transport
• shallow water equations.

References:

[1] E. Audusse et al., “A fast finite volume solver for multilayered shallow
water flows with mass exchange,” In: Journal of Computational Physics,
vol. 272, 2014, pp. 23-45.
[2] R. A. Bagnold, “An approach to the sediment transport problem
from general physics,” In: U.S. Geological Survey Professional Paper,
1 (1966).
[3] F. Benkhaldoun, S. Sari, and M. Seaid, “Projection finite volume method
for shallow water flows,” In: Mathematics and Computers in Simulation,
vol. 118, 2015, pp. 87-101.
[4] F. Benkhaldoun et al., “Comparison of unstructured finite-volume
morphodynamic models in contracting channel flows,” In: Mathematics
and Computers in Simulation, vol. 81, 2011, pp. 2081-2097.
[5] R. Briganti et al. , “An efficient and flexible solver for the simulation of
morphodynamics of fast evolving flows on coarse sediments beaches,”
In: International Journal for Numerical Methods in Fluids, vol. 69, 2011,
pp. 859-877.
[6] W. R. Brownlie, “Prediction of flow depth and sediment discharge in
open channels,” In: Report Number KH-R-43A Keck Laboratory of
Hydraulics and Water Resources, California Institute of Technology,
1981.
[7] Z. Cao and P. Carling, “Mathematical modelling of alluvial rivers: reality
and myth. Part I: General overview,” In: Water Maritime Engineering,
vol. 154, 2002, pp. 207-220.
[8] Z. Cao et al., “Computational dam-break hydraulics over erodible
sediment bed,” In: Journal of Hydraulic Engineering, vol. 67, 2004,
pp. 149-152.
[9] M. J. Castro Diaz, E. D. Fernadez-Nieto, and A. M. Ferreiro, “Sediment
transport models in Shallow Water equations and numerical approach
by high order finite volume methods,” In: Computers and Fluids, vol.
37, 2008, pp. 299-316.
[10] F. Engelund and J. Fredsoe, “A sediment transport model for straight
alluvial channels,” In: Nordic Hydrology, vol. 7, 1976, pp. 293-306.
[11] A. J. Grass, “Sediment Transport by Waves and Currents,” In: SERC
London Cent. Mar. Technol., FL29, 1981.
[12] M. Guan, N. G. Wright, and P. G. Sleigh, “Multimode morphodynamic
model for sediment-laden flows and geomorphic impacts,” In: Journal
of Hydraulic Engineering, vol. 141, 2015.
[13] J. Guo and P. Y. Julien, “Turbulent velocity profiles in sediment-laden
flows,” In: Journal of Hydraulic Research, vol. 39, 2001, pp. 11-23.
[14] S. Huang et al. , “Vertical distribution of sediment concentration,” In:
Journal of Zhejiang University SCIENCE, vol. 9, 2008, pp. 1560-1566.
[15] J. Hudson and P. K. Sweby, “Formations for numerically approximating
hyperbolic systems governing sediment transport,” In: Journal of
Scientific Computing, vol. 19, 2003, pp. 225-252.
[16] J. Hudson and et al., “Numerical approaches for 1D morphodynamic
modelling,” In: Coastal Eng, vol. 52, 2005, pp. 691-707.
[17] J. Hudson and et al., “Numerical modeling of sediment transport applied
to coastal morphodynamics,” In: Applied Numerical Mathematics, vol.
104, 2016, pp. 30-46.
[18] X. Lui, “New Near-Wall Treatment for Suspended Sediment Transport
Simulations with High Reynolds Number Turbulence Models,” In:
Journal of Hydraulic Engineering, vol. 140, 2014, pp. 333-339.
[19] X. Liu and A. Beljadid, “A coupled numerical model for water flow,
sediment transport and bed erosion,” In: Computers and Fluids, vol.
154, 2017, pp. 273-284.
[20] E. Meyer-Peter and R. Muller, “Formulas for Bed-load Transport,”
In: Report on 2nd meeting on international association on hydraulic
structures research, 1948, pp. 39-64.
[21] J. Murillo and P. Garcia-Navarro, “An Exner-based coupled model
for two-dimensional transient flow over erodible bed,” In: Journal of
Computational Physics, vol. 229, 2010, pp. 8704-8732.
[22] G. Rosatti and L. Fraccarollo, “A well-balanced approach for flows over
mobile-bed with high sediment transport,” In: Journal of Computational
Physics, vol. 220, 2006, pp. 312-338.
[23] W. W. Rubey, “Settling velocity of gravel, sand, and silt particles,” In:
American Journal of Science, vol. 148, 1933, pp. 325-338.
[24] S. Soares-Frazao and Y. Zech, “HLLC scheme with novel wave-speed
estimators appropriate for two dimensional shallow-water flow on
erodible bed,” In: International Journal for Numerical Methods in
Fluids, vol. 66, 2011, pp. 1019-1036.
[25] J. Thompson et al., Event-based total suspended sediment particle size
distribution model,” In: Journal of Hydrology, vol. 536, 2016, pp.
236-246.
[26] L. C. Van Rijn, “Sediment Pick up Functions,” In: Journal of Hydraulic
Engineering, vol. 110, 1984, pp. 1494-1502.
[27] L. C. Van Rijn, “Unified view of sediment transport by currents and
waves. I: Initiation of motion, bed roughness, and bed-load transport,”
In: Journal of Hydraulic Engineering, vol. 113, 2007, pp. 649-667.
[28] V. A. Vanoni and G. N. Nomicos, “Resistance Properties of
Sediment-Laden Streams,” In: Transactions of the American Society of
Civil Engineers, vol. 125, 1960, pp. 1140-1167.
[29] W. Wu and S. S. Wang, “Formulas for sediment porosity and settling
velocity,” In: Journal of Hydraulic Engineering, vol. 132, 2006, pp.
858-862.