26 research outputs found
A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model
The fully nonlinear and weakly dispersive Green-Naghdi model for shallow
water waves of large amplitude is studied. The original model is first recast
under a new formulation more suitable for numerical resolution. An hybrid
finite volume and finite difference splitting approach is then proposed. The
hyperbolic part of the equations is handled with a high-order finite volume
scheme allowing for breaking waves and dry areas. The dispersive part is
treated with a classical finite difference approach. Extensive numerical
validations are then performed in one horizontal dimension, relying both on
analytical solutions and experimental data. The results show that our approach
gives a good account of all the processes of wave transformation in coastal
areas: shoaling, wave breaking and run-up
Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model
We investigate here the ability of a Green-Naghdi model to reproduce strongly
nonlinear and dispersive wave propagation. We test in particular the behavior
of the new hybrid finite-volume and finite-difference splitting approach
recently developed by the authors and collaborators on the challenging
benchmark of waves propagating over a submerged bar. Such a configuration
requires a model with very good dispersive properties, because of the
high-order harmonics generated by topography-induced nonlinear interactions. We
thus depart from the aforementioned work and choose to use a new Green-Naghdi
system with improved frequency dispersion characteristics. The absence of dry
areas also allows us to improve the treatment of the hyperbolic part of the
equations. This leads to very satisfying results for the demanding benchmarks
under consideration
A Kinematic Conservation Law in Free Surface Flow
The Green-Naghdi system is used to model highly nonlinear weakly dispersive
waves propagating at the surface of a shallow layer of a perfect fluid. The
system has three associated conservation laws which describe the conservation
of mass, momentum, and energy due to the surface wave motion. In addition, the
system features a fourth conservation law which is the main focus of this note.
It is shown how this fourth conservation law can be interpreted in terms of a
concrete kinematic quantity connected to the evolution of the tangent velocity
at the free surface. The equation for the tangent velocity is first derived for
the full Euler equations in both two and three dimensional flows, and in both
cases, it gives rise to an approximate balance law in the Green-Naghdi theory
which turns out to be identical to the fourth conservation law for this system.Comment: 15 pages, 1 figur
A high-order spectral element unified boussinesq model for floating point absorbers
International audienceNonlinear wave-body problems are important in renewable energy, especially in case of wave energy converters operating in the near-shore region. In this paper we simulate nonlinear interaction between waves and truncated bodies using an efficient spectral/hp element depth-integrated unified Boussinesq model. The unified Boussinesq model treats also the fluid below the body in a depth-integrated approach. We illustrate the versatility of the model by predicting the reflection and transmission of solitary waves passing truncated bodies. We also use the model to simulate the motion of a latched heaving box. In both cases the unified Boussinesq model show acceptable agreement with CFD results-if applied within the underlying assumptions of dispersion and nonlinearity-but with a significant reduction in computational effort
A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes
In this paper, we introduce a discontinuous Finite Element formulation on
simplicial unstructured meshes for the study of free surface flows based on the
fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a
new class of asymptotically equivalent equations, which have a simplified
analytical structure, we consider a decoupling strategy: we approximate the
solutions of the classical shallow water equations supplemented with a source
term globally accounting for the non-hydrostatic effects and we show that this
source term can be computed through the resolution of scalar elliptic
second-order sub-problems. The assets of the proposed discrete formulation are:
(i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary
order of approximation in space, (iii) the exact preservation of the motionless
steady states, (iv) the preservation of the water height positivity, (v) a
simple way to enhance any numerical code based on the nonlinear shallow water
equations. The resulting numerical model is validated through several
benchmarks involving nonlinear wave transformations and run-up over complex
topographies