Saint-Venant Equations and Friction Law for Modelling Self-Channeling Granular Flows: From Analogue to Numerical Simulation

Consistent equations for turbulent open-channel flows on a smooth bottom are derived using a turbulence model of mixing length and an asymptotic expansion in two layers. A shallow-water scaling is used in an upper – or external – layer and a viscous scaling is used in a thin viscous – or internal – layer close to the bottom wall. A matching procedure is used to connect both expansions in an overlap domain. Depth-averaged equations are then obtained in the approximation of weakly sheared flows which is rigorously justified. We show that the Saint-Venant equations with a negligible deviation from a flat velocity profile and with a friction law are a consistent set of equations at a certain level of approximation. The obtained friction law is of the Kármán–Prandtl type and successfully compared to relevant experiments of the literature. At a higher precision level, a consistent three-equation model is obtained with the mathematical structure of the Euler equations of compressible fluids with relaxation source terms. This new set of equations includes shearing effects and adds corrective terms to the Saint-Venant model. At this level of approximation, energy and momentum resistances are clearly distinguished. Several applications of this new model that pertains to the hydraulics of open-channel flows are presented including the computation of backwater curves and the numerical resolution of the growing and breaking of roll waves.

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Numerical Simulation for Gas and Granular Flows in Raceway Region

The Proceedings of The Computational Mechanics Conference ◽

10.1299/jsmecmd.2004.17.739 ◽

2004 ◽

Vol 2004.17 (0) ◽

pp. 739-740

Author(s):

Toshihiko UMEKAGE ◽

Shinichi YUU

Keyword(s):

Numerical Simulation ◽

Granular Flows

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Numerical Simulation of Propagation and Run-Up of Long Waves in U-Shaped Bays

Fluids ◽

10.3390/fluids6040146 ◽

2021 ◽

Vol 6 (4) ◽

pp. 146

Author(s):

S. R. Pudjaprasetya ◽

Vania M. Risriani ◽

Iryanto

Keyword(s):

Numerical Simulation ◽

Wave Propagation ◽

Long Waves ◽

Staggered Grid ◽

Wave Focusing ◽

Consistent Approximation ◽

Saint Venant Equations ◽

Wave Shoaling ◽

Run Up ◽

Good Agreement

Wave propagation and run-up in U-shaped channel bays are studied here in the framework of the quasi-1D Saint-Venant equations. Our approach is numerical, using the momentum conserving staggered-grid (MCS) scheme, as a consistent approximation of the Saint-Venant equations. We carried out simulations regarding wave focusing and run-ups in U-shaped bays. We obtained good agreement with the existing analytical results on several aspects: the moving shoreline, wave shoaling, and run-up heights. Our findings also confirm that the run-up height is significantly higher in the parabolic bay than on a plane beach. This assessment shows the merit of the MCS scheme in describing wave focusing and run-up in U-shaped bays. Moreover, the MCS scheme is also efficient because it is based on the quasi-1D Saint-Venant equations.

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Nonuniform and Unsteady Sliding of a Plate Boundary in a Great Earthquake Cycle: A Numerical Simulation Using a Laboratory-derived Friction Law

Pure and Applied Geophysics ◽

10.1007/s000240050256 ◽

1999 ◽

Vol 155 (1) ◽

pp. 93-118 ◽

Cited By ~ 20

Author(s):

N. Kato ◽

T. Hirasawa

Keyword(s):

Numerical Simulation ◽

Plate Boundary ◽

Great Earthquake ◽

Earthquake Cycle ◽

Friction Law

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Numerical simulation of A-type hydraulic jumps at positive steps

Canadian Journal of Civil Engineering ◽

10.1139/l00-024 ◽

2000 ◽

Vol 27 (4) ◽

pp. 805-813 ◽

Cited By ~ 6

Author(s):

A Burcu Altan Sakarya ◽

Nuray Denli Tokyay

Keyword(s):

Numerical Simulation ◽

Hydraulic Jump ◽

Integral Approach ◽

Internal Boundary ◽

Subcritical Flow ◽

One Dimensional ◽

A Value ◽

Saint Venant Equations ◽

The One ◽

Positive Step

A numerical simulation of the A-type hydraulic jump at a positive step, which is an example of mixed supercritical-subcritical flow with a discontinuity at the channel bed, is given by using an integral approach. A gradually varied subcritical flow over a rectangular, horizontal, and prismatic channel with an abrupt bottom rise is considered as the initial condition. Then, the upstream depth is decreased to a value producing a supercritical flow and remaining unchanged during computations. The resulting unsteady flow is solved by using both the MacCormack and the dissipative two-four schemes for the one-dimensional, unsteady Saint-Venant equations. In the numerical simulation, the step is treated as an internal boundary. At the downstream and the internal boundaries, the method of characteristics is employed to compute the relevant parameters. The numerical simulation is verified by comparing the results with the available data and analytical methods.Key words: hydraulic jump, positive step, numerical simulation, internal boundary.

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