A kinematic conservation law in free surface flow

Gavrilyuk, Sergey; Kalisch, Henrik; Khorsand, Zahra

doi:10.1088/0951-7715/28/6/1805

Cited by 28 publications

(31 citation statements)

References 45 publications

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“…where the velocity potentials φ i (i = 1, 2) have been defined in Section 1.2, and the approximation is meant with precision O(μ 2 ). Thus, one has by construction d dt V i = O(μ 2 ), and it turns out that this approximately conserved quantity is actually exactly conserved by the Green-Naghdi flow (see also [37]). Of course, the linear combination (recall (8) and (9))…”

Section: Conserved Quantitiesmentioning

confidence: 97%

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

2016

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We introduce a new class of Green-Naghdi type models for the propagation of internal waves between two (1 + 1)-dimensional layers of homogeneous, immiscible, ideal, incompressible, irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi-layer Green-Naghdi model, and in particular to manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well-posedness and stability results. These results apply in particular to the original Green-Naghdi model as well as to the Saint-Venant (hydrostatic shallow-water) system with surface tension.

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Section: Conserved Quantitiesmentioning

confidence: 97%

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

2016

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“…Quite recently, equation (2.9) was generalized to the equation which is applicable to rotational flows (Castro & Lannes [20]). The latter equation was also derived in the framework of the Lagrangian formulation for the water wave problem (Gavrilyuk et al [21]). Equation (2.9) represents an exact conservation law for the vector V .…”

Section: (A) Extended Gn Systemmentioning

confidence: 99%

Hamiltonian structure for two-dimensional extended Green–Naghdi equations

Matsuno

2016

Proc. R. Soc. A.

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The two-dimensional Green-Naghdi (GN) shallow-water model for surface gravity waves is extended to incorporate the arbitrary higher-order dispersive effects. This can be achieved by developing a novel asymptotic analysis applied to the basic nonlinear water wave problem. The linear dispersion relation for the extended GN system is then explored in detail. In particular, we use its characteristics to discuss the well-posedness of the linearized problem. As illustrative examples of approximate model equations, we derive a higher-order model that is accurate to the fourth power of the dispersion parameter in the case of a flat bottom topography, and address the related issues such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom on the basis of this model. The original GN model is then briefly described in the case of an uneven bottom topography. Subsequently, the extended GN system presented here is shown to have the same Hamiltonian structure as that of the original GN system. Last, we demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

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“…Hence the pressure boundary condition can be expressed as a conservation law. In fact this is the conservation law of Gavrilyuk et al [8], hereafter the GKK conservation law. It is an exact conservation law at the free surface with no restriction on the velocity or vorticity field.…”

Section: The Pressure Boundary Conditionmentioning

confidence: 93%

The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves

Bridges

2019

Water Waves

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The pressure boundary condition for the full Euler equations with a free surface and general vorticity field is formulated in terms of a generalized Bernoulli equation deduced from the Gavrilyuk-Kalisch-Khorsand conservation law. The use of pressure as a Lagrangian density, as in Luke's variational principle, is reviewed and extension to a full vortical flow is attempted with limited success. However, a new variational principle for time-dependent water waves in terms of the stream function is found. The variational principle generates vortical boundary conditions but with a harmonic stream function. Other aspects of vorticity in variational principles are also discussed. Keywords Oceanography • Lagrangian • Vorticity • Streamfunction • Variational principle where (U , V) is the velocity field evaluated at the free surface, P a (x, t) is the imposed external pressure field, and f (t) is an arbitrary function of time. The function (x, t)

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A kinematic conservation law in free surface flow

Cited by 28 publications

References 45 publications

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

Hamiltonian structure for two-dimensional extended Green–Naghdi equations

The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves

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