scholarly journals On the strength of the weakly nonlinear theory for surface gravity waves

2016 ◽  
Vol 810 ◽  
pp. 1-4 ◽  
Author(s):  
Michael Stiassnie
Keyword(s):  
Gravity Waves ◽  
Nonlinear Theory ◽  
Mathematical Formulation ◽  
Wave Theory ◽  
Quantitative Agreement ◽  
Surface Gravity ◽  
Weakly Nonlinear ◽  
Surface Gravity Waves ◽  
Weakly Nonlinear Theory ◽  
Wave Basin

Recently, Bonnefoy et al. (J. Fluid Mech., vol. 805, 2016, R3) studied the resonant interaction of oblique surface gravity waves in a large $50~\text{m}\times 30~\text{m}\times 5~\text{m}$ wave basin. Their experimental results are in excellent quantitative agreement with predictions of the weakly nonlinear wave theory, and provide additional evidence to the strength of this widely used mathematical formulation. In this article, the reader is introduced to the many facets of the weakly nonlinear theory for surface gravity waves, and to its current and possible future applications, deterministic as well as stochastic.

scholarly journals The role of constant vorticity on weakly nonlinear surface gravity waves

Wave Motion ◽  
2020 ◽  
pp. 102702
Author(s):  
M.A. Manna ◽  
S. Noubissie ◽  
J. Touboul ◽  
B. Simon ◽  
R.A. Kraenkel
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Weakly Nonlinear ◽  
Surface Gravity Waves ◽  
Constant Vorticity ◽  
Nonlinear Surface

A rotating fluid cylinder subject to weak precession

2008 ◽  
Vol 599 ◽  
pp. 405-440 ◽  
Author(s):  
PATRICE MEUNIER ◽  
CHRISTOPHE ELOY ◽  
ROMAIN LAGRANGE ◽  
FRANÇOIS NADAL
Keyword(s):  
Reynolds Number ◽  
Nonlinear Theory ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Precession Angle ◽  
Weakly Nonlinear ◽  
Low Reynolds Numbers ◽  
Axisymmetric Mode ◽  
Mode Amplitude ◽  
Weakly Nonlinear Theory

In this paper, we report experimental and theoretical results on the flow inside a precessing and rotating cylinder. Particle image velocimetry measurements have revealed the instantaneous structure of the flow and confirmed that it is the sum of forced inertial (Kelvin) modes, as predicted by the classical linear inviscid theory. But this theory predicts also that the amplitude of a mode diverges when its natural frequency equals the precession frequency. A viscous and weakly nonlinear theory has therefore been developed at the resonance. This theory has been compared to experimental results and shows a good quantitative agreement. For low Reynolds numbers, the mode amplitude scales as the square root of the Reynolds number owing to the presence of Ekman layers on the cylinder walls. When the Reynolds number is increased, the amplitude saturates at a value which scales as the precession angle to the power one-third for a given resonance. The nonlinear theory also predicts the forcing of a geostrophic (axisymmetric) mode which has been observed and measured in the experiments. These results allow the flow inside a precessing cylinder to be fully characterized in all regimes as long as there is no instability.

scholarly journals On the Lagrangian description of steady surface gravity waves

2007 ◽  
Vol 589 ◽  
pp. 433-454 ◽  
Author(s):  
DIDIER CLAMOND
Keyword(s):  
Gravity Waves ◽  
Order Approximation ◽  
Mathematical Formulation ◽  
Finite Depth ◽  
Stokes Drift ◽  
Surface Gravity ◽  
Lagrangian Description ◽  
Perturbation Scheme ◽  
Third Order ◽  
Surface Gravity Waves

This paper concerns the mathematical formulation of two-dimensional steady surface gravity waves in a Lagrangian description of motion. It is demonstrated first that classical second-order Lagrangian Stokes-like approximations do not exactly represent a steady wave motion in the presence of net mass transport (Stokes drift). A general mathematically correct formulation is then derived. This derivation leads naturally to a Lagrangian Stokes-like perturbation scheme that is uniformly valid for all time – in other words, without secular terms. This scheme is illustrated, both for irrotational waves, with seventh-order and third-order approximations in deep water and finite depth, respectively, and for rotational waves with a third-order approximation of the Gerstner-like wave on finite depth. It is also shown that the Lagrangian approximations are more accurate than their Eulerian counterparts of the same order.

scholarly journals Effects of an abrupt depth change on weakly nonlinear surface gravity waves: deterministic and stochastic analysis

2020 ◽  
Author(s):  
Yan Li ◽  
Samuel Draycott ◽  
Yaokun Zheng ◽  
Thomas A.A. Adcock ◽  
Zhiliang Lin ◽  
...  
Keyword(s):  
Gravity Waves ◽  
Second Order ◽  
Surface Gravity ◽  
Irregular Waves ◽  
Wave Steepness ◽  
Weakly Nonlinear ◽  
Wave Statistics ◽  
Surface Gravity Waves ◽  
Seabed Topography ◽  
Nonlinear Surface

<p>This work focuses on two different aspects of the effect of an abrupt depth transition on weakly nonlinear surface gravity waves: deterministic and stochastic. It is known that the kurtosis of waves can reach a maximum near the top of such abrupt depth transitions. The analysis is based on three different approaches: (1) a novel theoretical framework that allows for narrow-banded surface waves experiencing a step-type seabed, correct to the second order in wave steepness; (2) experimental observations; and (3) a numerical model based on a fully nonlinear potential flow solver. To reveal the fundamental physics, the evolution of a wave envelope that experiences an abrupt depth transition is examined in detail; (a) we show the release of free waves at second order in wave steepness both for the super-harmonic and sub-harmonic or ‘mean’ terms; (b) a local wave height peak that occurs near the top of a depth transition – whose exact position depends on several nondimensional parameters – is revealed; (c) furthermore, we examine which parameters affect this peak. The novel physics has implications for wave statistics for long-crested irregular waves experiencing an abrupt depth transition. We show the connection of the second-order physics at work in the deterministic and stochastic cases: the peak of wave kurtosis and skewness occurs in the neighborhood of the deterministic wave peak in (b) and for the same parameters set composed of a seabed topography, water depths, primary wave frequency and steepness, and bandwidth.</p>

scholarly journals Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a three-dimensional wave basin

2009 ◽  
Vol 627 ◽  
pp. 235-257 ◽  
Author(s):  
M. ONORATO ◽  
L. CAVALERI ◽  
S. FOUQUES ◽  
O. GRAMSTAD ◽  
P. A. E. M. JANSSEN ◽  
...  
Keyword(s):  
Gravity Waves ◽  
Three Dimensional ◽  
Order Theory ◽  
Second Order ◽  
Statistical Properties ◽  
Surface Elevation ◽  
Surface Gravity ◽  
Surface Gravity Waves ◽  
Wave Basin ◽  
Dimensional Wave

A wave basin experiment has been performed in the MARINTEK laboratories, in one of the largest existing three-dimensional wave tanks in the world. The aim of the experiment is to investigate the effects of directional energy distribution on the statistical properties of surface gravity waves. Different degrees of directionality have been considered, starting from long-crested waves up to directional distributions with a spread of ±30° at the spectral peak. Particular attention is given to the tails of the distribution function of the surface elevation, wave heights and wave crests. Comparison with a simplified model based on second-order theory is reported. The results show that for long-crested, steep and narrow-banded waves, the second-order theory underestimates the probability of occurrence of large waves. As directional effects are included, the departure from second-order theory becomes less accentuated and the surface elevation is characterized by weak deviations from Gaussian statistics.

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