9,058 research outputs found
Wave modelling - the state of the art
This paper is the product of the wave modelling community and it tries to make a picture of the present situation in this branch of science, exploring the previous and the most recent results and looking ahead towards the solution of the problems we presently face. Both theory and applications are considered.
The many faces of the subject imply separate discussions. This is reflected into the single sections, seven of them, each dealing with a specific topic, the whole providing a broad and solid overview of the present state of the art. After an introduction framing the problem and the approach we followed, we deal in sequence with the following subjects: (Section) 2, generation by wind; 3, nonlinear interactions in deep water; 4, white-capping dissipation; 5, nonlinear interactions in shallow water; 6, dissipation at the sea bottom; 7, wave propagation; 8, numerics. The two final sections, 9 and 10, summarize the present situation from a general point of view and try to look at the future developments
Current effects on scattering of surface gravity waves by bottom topography
Scattering of random surface gravity waves by small amplitude topography in
the presence of a uniform current is investigated theoretically. This problem
is relevant to ocean waves propagation on shallow continental shelves where
tidal currents are often significant. A perturbation expansion of the wave
action to second order in powers of the bottom amplitude yields an evolution
equation for the wave action spectrum. A scattering source term gives the rate
of exchange of the wave action spectrum between wave components, with
conservation of the total action at each absolute frequency. With and without
current, the scattering term yields reflection coefficients for the amplitudes
of waves that converge, to the results of previous theories for monochromatic
waves propagating in one dimension over sinusoidal bars. Over sandy continental
shelves, tidal currents are known to generate sandwaves with scales comparable
to those of surface waves. Application of the theory to such a real topography
suggests that scattering mainly results in a broadening of the directional wave
spectrum, due to forward scattering, while the back-scattering is generally
weaker. The current may strongly influence surface gravity wave scattering by
selecting different bottom scales with widely different spectral densities due
the sharp bottom spectrum roll-off.Comment: submitted to Journal of Fluid Mechanics 7 Oct 200
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
We design an arbitrary high-order accurate nodal discontinuous Galerkin
spectral element approximation for the nonlinear two dimensional shallow water
equations with non-constant, possibly discontinuous, bathymetry on
unstructured, possibly curved, quadrilateral meshes. The scheme is derived from
an equivalent flux differencing formulation of the split form of the equations.
We prove that this discretisation exactly preserves the local mass and
momentum. Furthermore, combined with a special numerical interface flux
function, the method exactly preserves the mathematical entropy, which is the
total energy for the shallow water equations. By adding a specific form of
interface dissipation to the baseline entropy conserving scheme we create a
provably entropy stable scheme. That is, the numerical scheme discretely
satisfies the second law of thermodynamics. Finally, with a particular
discretisation of the bathymetry source term we prove that the numerical
approximation is well-balanced. We provide numerical examples that verify the
theoretical findings and furthermore provide an application of the scheme for a
partial break of a curved dam test problem
Experimental study of breathers and rogue waves generated by random waves over non-uniform bathymetry
Experimental results describing random, uni-directional, long crested, water
waves over non-uniform bathymetry confirm the formation of stable coherent wave
packages traveling with almost uniform group velocity. The waves are generated
with JONSWAP spectrum for various steepness, height and constant period. A set
of statistical procedures were applied to the experimental data, including the
space and time variation of kurtosis, skewness, BFI, Fourier and moving Fourier
spectra, and probability distribution of wave heights. Stable wave packages
formed out of the random field and traveling over shoals, valleys and slopes
were compared with exact solutions of the NLS equation resulting in good
matches and demonstrating that these packages are very similar to deep water
breathers solutions, surviving over the non-uniform bathymetry. We also present
events of formation of rogue waves over those regions where the BFI, kurtosis
and skewness coefficients have maximal values.Comment: 41 pages, 21 figure
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
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