313 research outputs found
Traveling surface waves of moderate amplitude in shallow water
We study traveling wave solutions of an equation for surface waves of
moderate amplitude arising as a shallow water approximation of the Euler
equations for inviscid, incompressible and homogenous fluids. We obtain
solitary waves of elevation and depression, including a family of solitary
waves with compact support, where the amplitude may increase or decrease with
respect to the wave speed. Our approach is based on techniques from dynamical
systems and relies on a reformulation of the evolution equation as an
autonomous Hamiltonian system which facilitates an explicit expression for
bounded orbits in the phase plane to establish existence of the corresponding
periodic and solitary traveling wave solutions
Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I: General presentation and periodic solutions
We present experimental results on hydrothermal traveling-waves dynamics in
long and narrow 1D channels. The onset of primary traveling-wave patterns is
briefly presented for different fluid heights and for annular or bounded
channels, i.e., within periodic or non-periodic boundary conditions. For
periodic boundary conditions, by increasing the control parameter or changing
the discrete mean-wavenumber of the waves, we produce modulated waves patterns.
These patterns range from stable periodic phase-solutions, due to supercritical
Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes
and/or counter-propagating-waves competition, i.e., traveling sources and
sinks. The transition from non-linearly saturated Eckhaus modulations to
transient pattern-breaks by traveling holes and spatio-temporal defects is
documented. Our observations are presented in the framework of coupled complex
Ginzburg-Landau equations with additional fourth and fifth order terms which
account for the reflection symmetry breaking at high wave-amplitude far from
onset. The second part of this paper (nlin.PS/0208030) extends this study to
spatially non-periodic patterns observed in both annular and bounded channel.Comment: 45 pages, 21 figures (elsart.cls + AMS extensions). Accepted in
Physica D. See also companion paper "Nonlinear dynamics of waves and
modulated waves in 1D thermocapillary flows. II: Convective/absolute
transitions" (nlin.PS/0208030). A version with high resolution figures is
available on N.G. web pag
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
2009 program of studies : nonlinear waves
The fiftieth year of the program was dedicated to Nonlinear Waves, a topic with many
applications in geophysical fluid dynamics. The principal lectures were given jointly by
Roger Grimshaw and Harvey Segur and between them they covered material drawn from
fundamental theory, fluid experiments, asymptotics, and reaching all the way to detailed
applications. These lectures set the scene for the rest of the summer, with subsequent
daily lectures by staff and visitors on a wide range of topics in GFD. It was a challenge
for the fellows and lecturers to provide a consistent set of lecture notes for such a wide-ranging
lecture course, but not least due to the valiant efforts of Pascale Garaud, who
coordinated the write-up and proof-read all the notes, we are very pleased with the final
outcome contained in these pages.
This yearâs group of eleven international GFD fellows was as diverse as one could get in
terms of gender, origin, and race, but all were unified in their desire to apply their
fundamental knowledge of fluid dynamics to challenging problems in the real world.
Their projects covered a huge range of physical topics and at the end of the summer each
student presented his or her work in a one-hour lecture. As always, these projects are the
heart of the research and education aspects of our summer study.Funding was provided by the National Science Foundation through Grant No. OCE-0824636 and
the Office of Naval Research under Contract No. N00014-09-10844
Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
International audienceWe study the spectral stability of roll-wave solutions of the viscous St. Venant equationsmodeling inclined shallow-water flow, both at onset in the small-Froude number or âweakly unstableâlimit F â 2+ and for general values of the Froude number F , including the limit F â +â. In the former,F â 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV)equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate thisformal limit, showing that stability as F â 2+ is equivalent to stability of the corresponding KdV-KSwaves in the KdV limit. Together with recent results obtained for KdV-KS by JohnsonâNobleâRodriguesâZumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St.Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainderof the paper, we investigate numerically and analytically the evolution of the stability diagram as Froudenumber increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable todifferent, large-F behavior, with stability determined by simple power law relations. The latter stabilitycriteria are potentially useful in hydraulic engineering applications, for which typically 2.5 †F †6.0
- âŠ