268 research outputs found
Evolution of Random Wave Fields in the Water of Finite Depth
The evolution of random wave fields on the free surface is a complex process
which is not completely understood nowadays. For the sake of simplicity in this
study we will restrict our attention to the 2D physical problems only (i.e. 1D
wave propagation). However, the full Euler equations are solved numerically in
order to predict the wave field dynamics. We will consider the most studied
deep water case along with several finite depths (from deep to shallow waters)
to make a comparison. For each depth we will perform a series of Monte--Carlo
runs of random initial conditions in order to deduce some statistical
properties of an average sea state.Comment: 12 pages, 5 figures, 28 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Group and phase velocities in the free-surface visco-potential flow: new kind of boundary layer induced instability
Water wave propagation can be attenuated by various physical mechanisms. One
of the main sources of wave energy dissipation lies in boundary layers. The
present work is entirely devoted to thorough analysis of the dispersion
relation of the novel visco-potential formulation. Namely, in this study we
relax all assumptions of the weak dependence of the wave frequency on time. As
a result, we have to deal with complex integro-differential equations that
describe transient behaviour of the phase and group velocities. Using numerical
computations, we show several snapshots of these important quantities at
different times as functions of the wave number. Good qualitative agreement
with previous study [Dutykh2009] is obtained. Thus, we validate in some sense
approximations made anteriorly. There is an unexpected conclusion of this
study. According to our computations, the bottom boundary layer creates
disintegrating modes in the group velocity. In the same time, the imaginary
part of the phase velocity remains negative for all times. This result can be
interpreted as a new kind of instability which is induced by the bottom
boundary layer effect.Comment: 12 pages, 7 figures. Reviewer's comments were taken into account.
Other author's papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutykh
Hamiltonian description and traveling waves of the spatial Dysthe equations
The spatial version of the fourth-order Dysthe equations describe the
evolution of weakly nonlinear narrowband wave trains in deep waters. For
unidirectional waves, the hidden Hamiltonian structure and new invariants are
unveiled by means of a gauge transformation to a new canonical form of the
evolution equations. A highly accurate Fourier-type spectral scheme is
developed to solve for the equations and validate the new conservation laws,
which are satisfied up to machine precision. Further, traveling waves are
numerically investigated using the Petviashvili method. It is found that their
collision appears inelastic, suggesting the non-integrability of the Dysthe
equations.Comment: Research report. 17 pages, 7 figures, 38 references. Other author's
papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/ . arXiv
admin note: substantial text overlap with arXiv:1110.408
Camassa-Holm type equations for axisymmetric Poiseuille pipe flows
We present a study on the nonlinear dynamics of a disturbance to the laminar
state in non-rotating axisymmetric Poiseuille pipe flows. The associated
Navier-Stokes equations are reduced to a set of coupled generalized
Camassa-Holm type equations. These support singular inviscid travelling waves
with wedge-type singularities, the so called peakons, which bifurcate from
smooth solitary waves as their celerity increase. In physical space they
correspond to localized toroidal vortices or vortexons. The inviscid vortexon
is similar to the nonlinear neutral structures found by Walton (2011) and it
may be a precursor to puffs and slugs observed at transition, since most likely
it is unstable to non-axisymmetric disturbances.Comment: 11 pages, 4 figures, 31 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Resonance enhancement by suitably chosen frequency detuning
In this Letter we report new effects of resonance detuning on various
dynamical parameters of a generic 3-wave system. Namely, for suitably chosen
values of detuning the variation range of amplitudes can be significantly wider
than for exact resonance. Moreover, the range of energy variation is not
symmetric with respect to the sign of the detuning. Finally, the period of the
energy oscillation exhibits non-monotonic dependency on the magnitude of
detuning. These results have important theoretical implications where nonlinear
resonance analysis is involved, such as geophysics, plasma physics, fluid
dynamics. Numerous practical applications are envisageable e.g. in energy
harvesting systems.Comment: 13 pages, 6 figures, 5 references. Other author's papers can be
viewed at http://www.denys-dutykh.com
On supraconvergence phenomenon for second order centered finite differences on non-uniform grids
In the present study we consider an example of a boundary value problem for a
simple second order ordinary differential equation, which may exhibit a
boundary layer phenomenon. We show that usual central finite differences, which
are second order accurate on a uniform grid, can be substantially upgraded to
the fourth order by a suitable choice of the underlying non-uniform grid. This
example is quite pedagogical and may give some ideas for more complex problems.Comment: 26 pages, 2 figures, 2 tables, 37 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Dissipative Boussinesq equations
The classical theory of water waves is based on the theory of inviscid flows.
However it is important to include viscous effects in some applications. Two
models are proposed to add dissipative effects in the context of the Boussinesq
equations, which include the effects of weak dispersion and nonlinearity in a
shallow water framework. The dissipative Boussinesq equations are then
integrated numerically.Comment: 40 pages, 15 figures, published in C. R. Mecanique 335 (2007) Other
author's papers can be downloaded at http://www.cmla.ens-cachan.fr/~dutyk
Multi-symplectic structure of fully-nonlinear weakly-dispersive internal gravity waves
In this short communication we present the multi-symplectic structure for the
two-layer Serre-Green-Naghdi equations describing the evolution of large
amplitude internal gravity long waves. We consider only a two-layer
stratification with rigid bottom and lid for simplicity, generalisations to
several layers being straightforward. This multi-symplectic formulation allows
the application of various multi-symplectic integrators (such as Euler or
Preissman box schemes) that preserve exactly the multi-symplecticity at the
discrete level.Comment: 15 pages, 1 figure, 15 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
On the relevance of the dam break problem in the context of nonlinear shallow water equations
The classical dam break problem has become the de facto standard in
validating the Nonlinear Shallow Water Equations (NSWE) solvers. Moreover, the
NSWE are widely used for flooding simulations. While applied mathematics
community is essentially focused on developing new numerical schemes, we tried
to examine the validity of the mathematical model under consideration. The main
purpose of this study is to check the pertinence of the NSWE for flooding
processes. From the mathematical point of view, the answer is not obvious since
all derivation procedures assumes the total water depth positivity. We
performed a comparison between the two-fluid Navier-Stokes simulations and the
NSWE solved analytically and numerically. Several conclusions are drawn out and
perspectives for future research are outlined.Comment: 20 pages, 15 figures. Accepted to Discrete and Continuous Dynamical
Systems. Other author's papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutyk
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