Two-layer ?Shallow water? model

Ovsyannikov, L. V.

doi:10.1007/bf00910010

Cited by 88 publications

(107 citation statements)

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“…In this paper we apply our relaxation technique to the non-conservative two-phase model (8). The same idea could be analogously used for other non-conservative systems, such as the two-layer shallow flow model [47][48][49]40,50]. Nonetheless, we have to remind the potential difficulties of the method related to the computation of non-conservative products, see Remark 7.1.…”

Section: Equivalence With the Vfroe Solvermentioning

confidence: 99%

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Pelanti

Bouchut

Mangeney

2011

Journal of Computational Physics

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a b s t r a c tWe present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resulting relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe's method, while maintaining Roe's low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions.As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallouët and Masella [T. Gallouët, J.-M. Masella, Un schéma de Godunov approché C.R. Acad. Sci. Paris, Série I, 323 (1996) 77-84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.

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Section: Equivalence With the Vfroe Solvermentioning

confidence: 99%

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Pelanti

Bouchut

Mangeney

2011

Journal of Computational Physics

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“…The Eulerian form of the shallow-water mass and momentum equations in each layer in the body frame are 5) where g > 0 is the gravitational acceleration constant, p s (x 2 , t) is the unknown pressure at the rigid lid, and the over dots denote a full derivative with respect to t [13,16]. The fluid in each layer must satisfy the no-penetration conditions on the vessel side walls, and hence, the boundary conditions are…”

Section: Coupled Equations With a Two-layer Fluidmentioning

confidence: 99%

“…The literature on two-layer flows in open systems, with and without a rigid lid is vast ( [5][6][7][8] to name a few), but in closed sloshing systems the literature is much more limited. The theoretical and experimental works of [9,10] show excellent agreement for sloshing in a fixed rectangular tank with a rigid lid when a Lagrangian representation of the system is reduced to a system of ordinary differential equations with dissipative damping.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion

2017

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The coupled motion-between multiple inviscid, incompressible, immiscible fluid layers in a rectangular vessel with a rigid lid and the vessel dynamics-is considered. The fluid layers are assumed to be thin and the shallowwater assumption is applied. The governing form of the Lagrangian functional in the Lagrangian particle path (LPP) framework is derived for an arbitrary number of layers, while the corresponding Hamiltonian is explicitly derived in the case of two-and three-layer fluids. The Hamiltonian formulation has nice properties for numerical simulations, and a fast, effective and symplectic numerical scheme is presented in the two-and three-layer cases, based upon the implicit-midpoint rule. Results of the simulations are compared with linear solutions and with the existing results of Alemi Ardakani et al. (J Fluid Struct 59:432-460, 2015) which were obtained using a finite volume approach in the Eulerian representation. The latter results are extended to non-Boussinesq regimes. The advantages and limitations of the LPP formulation and variational discretization are highlighted.

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“…Different models can be derived, depending on the closure that is chosen for the definition of the layer, see [2,21]. Other multilayer models was proposed in the literature for stratified flow [17,20] but does not enter in the scope of this work. In the layerwise discretization [4] that we consider here, the layer thickness is assumed to be proportional to the total water depth.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

Aguillon¹,

Audusse²,

Godlewski³

et al. 2018

SIAM J. Math. Anal.

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Some shallow water type models describing the vertical profile of the horizontal velocity with several degrees of freedom have been recently proposed. The question addressed in the current work is the hyperbolicity of a shallow water model with two velocities. The model is written in a nonconservative form and the analysis of its eigenstructure shows the possibility that two eigenvalues coincide. A definition of the nonconservative product is given which enables us to analyse the resonance and coalescence of waves. Eventually, we prove the well-posedness of the two dimensional Riemann problem with initial condition constant by half-plane.

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Two-layer ?Shallow water? model

Cited by 88 publications

References 1 publication

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

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