Shallow water asymptotic models for the propagation of internal waves

Duchêne, Vincent; Israwi, Samer; Talhouk, Raafat

doi:10.3934/dcdss.2014.7.239

Cited by 10 publications

(21 citation statements)

References 49 publications

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“…Performing these computations, with extensive testing of their numerical accuracy (see also [2,10]), demonstrated a remarkable robustness of the nondispersive model predictions, at least for the class of initial conditions and time scales of evolution we have focussed on, at both qualitative and, with few exceptions, quantitative levels. This is borne out of a comparison between interface profiles obtained through the numerical solutions of the two-layer model (4.3) and the direct numerical simulation of their Euler counterpart by overlaying the profile on the late time snapshots from figures 14,15,18,19, as reported in figure 20. A notable example that emerges from this comparison is the prediction of "wing" structures (simple waves for the hyperbolic models) and their locations.…”

Section: Discussionmentioning

confidence: 99%

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

et al. 2018

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Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single layer homogeneous fluid with a constant pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These demonstrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results. arXiv:1812.02963v1 [physics.flu-dyn]

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Section: Discussionmentioning

confidence: 99%

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

et al. 2018

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“…In this section, we briefly recall the derivation of the full Euler system governing the evolution equations of the two-layers flow and refer to [1,5,16,17] for more details. The two-layers flow considered are assumed to be incompressible, homogeneous, immiscible perfect fluids of different densities under the sole influence of gravity.…”

Section: Full Euler Systemmentioning

confidence: 99%

“…We refer the reader to the following papers [2,9,30,6,29]. Earlier works have also set a very interesting theoretical background for the two-fluid system see [5,1,16,17,18], for more details. One of these reduced models is the Green-Naghdi system of partial differential equations (denoted GN in the following).…”

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confidence: 99%

A numerical scheme for an improved Green–Naghdi model in the Camassa–Holm regime for the propagation of internal waves

2017

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“…where we used Lemma A.4. Of course, one could have simply kept the surface tension term unchanged at this point, as in [20]. The smallness of surface tension, expressed by bo −1 ≤ bo −1 min , is useful in the derivation of our new model, in the following subsection.…”

Section: The Green-naghdi Modelmentioning

confidence: 99%

A New Fully Justified Asymptotic Model for the Propagation of Internal Waves in the Camassa--Holm Regime

Duchêne¹,

Israwi²,

Talhouk³

2015

SIAM J. Math. Anal.

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This study deals with asymptotic models for the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. We present a new Green-Naghdi type model in the Camassa-Holm (or medium amplitude) regime. This model is fully justified, in the sense that it is consistent, well-posed, and that its solutions remain close to exact solutions of the full Euler system with corresponding initial data. Moreover, our system allows to fully justify any well-posed and consistent lower order model; and in particular the so-called Constantin-Lannes approximation, which extends the classical Korteweg-de Vries equation in the Camassa-Holm regime. (convergence) is the difference between these two solutions small over the relevant time scale?As mentioned earlier, Lannes has recently proved [29] that the Cauchy problem for bi-fluidic full Euler system is well-posed in Sobolev spaces, in the presence of a small amount of surface tension. Thus the full justification of a consistent system of equation as an asymptotic model, in the sense described above, follows from its well-posedness and a stability result; see [27, Appendix C] for a detailed discussion and state of the art in the water-wave setting.A striking discrepancy between the water-wave and the bi-fluidic setting is that in the latter, large amplitude internal waves are known to generate Kelvin-Helmholtz instabilities, so that surface tension is necessary in order to regularize the flow. A crucial contribution of [29] consists in asserting that "the Kelvin-Helmholtz instabilities appear above a frequency threshold for which surface tension is relevant, while the main (observable) part of the wave involves low frequencies located below this frequency threshold". It is therefore expected that the surface tension does not play an essential role in the dominant evolution of the flow, especially in the shallow water regime. This intuition is confirmed by the fact that well-posedness and stability results have been proved for the bi-fluidic shallow-water system [22], and a class of Boussinesq-type systems [18], without surface tension and under reasonable assumptions on the flow (typically, the shear velocity must be sufficiently small). However, the original bi-fluidic Green-Naghdi model is known to be unconditionally ill-posed [30], which has led to various propositions in order to overcome this difficulty; see [3,13] and references therein. Let us recall here that Green-Naghdi models consist in higher order extensions of the shallow water equation, thus are consistent with precision O(µ 2 ) instead of O(µ), and allow strong nonlinearities (whereas Boussinesq models are limited to the long wave regime: = O(µ)). Finally, we mention the work of Xu [39], which studies and rigorously justifies the so-called intermediate long wave system, obtained in a regime similar to ours: ∼ √ µ, but δ ∼ √ µ.

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Shallow water asymptotic models for the propagation of internal waves

Cited by 10 publications

References 49 publications

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

A numerical scheme for an improved Green–Naghdi model in the Camassa–Holm regime for the propagation of internal waves

A New Fully Justified Asymptotic Model for the Propagation of Internal Waves in the Camassa--Holm Regime

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