On the Hyperbolicity of Two- and Three-Layer Shallow Water Equations

Castro, Manuel J.; Frings, Jörn Thies; Noelle, Sebastian; Parés, Carlos; Puppo, Gabriella

doi:10.1142/9789814417099_0030

Cited by 14 publications

(14 citation statements)

References 6 publications

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“…Accordingly, a special treatment has to be developed. The introduction of an intermediate third layer seems viable for recovering hyperbolicity (Castro et al 2010). Nevertheless, Eqs.…”

Section: Numerical Algorithmmentioning

confidence: 99%

A double layer-averaged model for dam-break flows over mobile bed

Cao

Pender

et al. 2013

Journal of Hydraulic Research

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Dam-break flows over mobile bed are often sharply stratified, comprising a bedload sediment-laden layer and an upper clear-water layer. Double layer-averaged (DL) models are attractive for modelling such flows due to the balance between the computing cost and the ability to represent stratification. However, existing DL models are oversimplified as sediment concentration in the sediment-laden layer is presumed constant, which is not generally justified. Here a new DL model is presented, explicitly incorporating the sediment mass conservation law in lieu of the assumption of constant sediment concentration. The two hyperbolic systems of the governing equations for the two layers are solved separately and simultaneously. The new model is demonstrated to agree with the experimental measurements of instant and progressive dam-break floods better than a simplified double layer-averaged model and a single layer-averaged model. It shows promise for applications to sharply stratified sediment-laden flows over mobile bed.

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“…Accordingly, a special treatment has to be developed. The introduction of an intermediate third layer seems viable for recovering hyperbolicity (Castro et al 2010). Nevertheless, Eqs.…”

Section: Numerical Algorithmmentioning

confidence: 99%

A double layer-averaged model for dam-break flows over mobile bed

Cao

Pender

et al. 2013

Journal of Hydraulic Research

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“…For instance, the reader is referred to works devoted to the well-known shallow-water system [5][6][7][8] and some related model [9][10][11][12] (see also [13] for isentropic steady states associated with (1)). For instance, the reader is referred to works devoted to the well-known shallow-water system [5][6][7][8] and some related model [9][10][11][12] (see also [13] for isentropic steady states associated with (1)).…”

Section: Lemmamentioning

confidence: 99%

“…This PDE system can be easily solved as soon as the pressure law only depends on the density. For instance, the reader is referred to works devoted to the well-known shallow-water system [5][6][7][8] and some related model [9][10][11][12] (see also [13] for isentropic steady states associated with (1)). Unfortunately, because the pressure function p depends on both and e, we cannot exhibit algebraic relations satisfied by the solutions of (14), except under restrictive assumptions.…”

Section: Lemmamentioning

confidence: 99%

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A well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity

Desveaux

Zenk

Berthon

et al. 2015

Numerical Methods in Fluids

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Summary This paper describes a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear partial differential equation, whose solutions are called hydrostatic equilibria. We present a well‐balanced method, meaning that besides discretizing the complete equations, the method is also able to maintain all hydrostatic equilibria. The method is a finite volume method, whose Riemann solver is approximated by a so‐called relaxation Riemann solution that takes all hydrostatic equilibria into account. Relaxation ensures robustness, accuracy, and stability of our method, because it satisfies discrete entropy inequalities. We will present numerical examples, illustrating that our method works as promised. Copyright © 2015 John Wiley & Sons, Ltd.

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“…(1)- (5) constitute a nonlinear system of eight partial differential equations, involving more equations than previously analyzed (e.g., Kim and LeVeque 2008;Castro et al 2010). It is too complicated to be solved numerically as a single system presently, which is reserved for future studies.…”

Section: Numerical Algorithmmentioning

confidence: 99%

Whole-Process Modeling of Reservoir Turbidity Currents by a Double Layer-Averaged Model

Cao

Pender

et al. 2015

J. Hydraul. Eng.

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Turbidity current is formed as subaerial open-channel sediment-laden flow plunges into a reservoir. The whole process of reservoir turbidity current, i.e., formation, propagation, and recession, is generally controlled by the water and sediment inputs from upstream and also the reservoir operation scheme specifying the downstream boundary condition. Enhanced understanding of reservoir turbidity current is critical to effective sediment management in alluvial rivers. However, until now there has been a lack of physically based and practically feasible models for resolving the whole process of reservoir turbidity current. This is because the computing cost of three-dimensional modeling is excessively high. Also, single layer-averaged models cannot resolve the formation process characterized by the transition from open-channel sediment-laden flow to subaqueous turbidity current, or the upper clear-water flow as dictated by the operation scheme of the reservoir, which has significant impacts on turbidity current. Here a new two-dimensional double layer-averaged model is proposed to facilitate for the first time whole-process modeling of reservoir turbidity current. The two hyperbolic systems of the governing equations for the two layers are solved separately and synchronously. The model is well balanced because the interlayer interactions are negligible compared with inertia and gravitation, featuring a reasonable balance between the flux gradients and the bed or interface slope source terms and thus applicable to irregular topographies. The model is benchmarked against a spectrum of experimental cases, including turbidity currents attributable to lock-exchange and sustained inflow. It is revealed that an appropriate clear-water outflow is favorable for turbidity current propagation and conducive to improving sediment flushing efficiency. This is significant for optimizing reservoir operation schemes. As applied to turbidity current in the Xiaolangdi Reservoir in the Yellow River, China, the model successfully resolves the whole process from formation to recession. The present work facilitates a viable and promising framework for whole-process modeling of turbidity currents, in support of reservoir sediment management.

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On the Hyperbolicity of Two- and Three-Layer Shallow Water Equations

Cited by 14 publications

References 6 publications

A double layer-averaged model for dam-break flows over mobile bed

A double layer-averaged model for dam-break flows over mobile bed

A well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity

Whole-Process Modeling of Reservoir Turbidity Currents by a Double Layer-Averaged Model

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