scholarly journals Large-Amplitude Steady Downstream Water Waves

2021 ◽  
Author(s):  
Adrian Constantin ◽  
Walter Strauss ◽  
Eugen Vărvărucă
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Large Amplitude ◽  
Water Surface ◽  
Maximum Amplitude ◽  
Uniform Bounds ◽  
Water Flows ◽  
Constant Vorticity ◽  
The Waves ◽  
Uniformly Bounded

AbstractWe study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed. For large-amplitude periodic traveling gravity waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are uniformly bounded.

scholarly journals Kelvin wake pattern at large Froude numbers

2013 ◽  
Vol 738 ◽  
Author(s):  
Alexandre Darmon ◽  
Michael Benzaquen ◽  
Elie Raphaël
Keyword(s):  
Gravity Waves ◽  
Water Surface ◽  
Pressure Field ◽  
Maximum Amplitude ◽  
Specific Pattern ◽  
Surface Form ◽  
Wake Region ◽  
Strong Hypothesis ◽  
Lord Kelvin ◽  
The Waves

AbstractGravity waves generated by an object moving at constant speed at the water surface form a specific pattern commonly known as the Kelvin wake. It was proved by Lord Kelvin that such a wake is delimited by a constant angle ${\simeq }19. 4{7}^{\circ } $. However a recent study by Rabaud and Moisy based on the observation of airborne images showed that the wake angle seems to decrease as the Froude number $Fr$ increases, scaling as $F{r}^{- 1} $ for large Froude numbers. To explain such observations they make the strong hypothesis that an object of size $b$ cannot generate wavelengths larger than $b$. Without the need of such an assumption and modelling the moving object by an axisymmetric pressure field, we analytically show that the angle corresponding to the maximum amplitude of the waves scales as $F{r}^{- 1} $ for large Froude numbers, whereas the angle delimiting the wake region outside which the surface is essentially flat remains constant and equal to the Kelvin angle for all $Fr$.

The interaction of large amplitude shallow-water waves with an ambient shear flow: non-critical flows

1972 ◽  
Vol 56 (2) ◽  
pp. 241-255 ◽  
Author(s):  
P. A. Blythe ◽  
Y. Kazakia ◽  
E. Varley
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Large Amplitude ◽  
Vertical Direction ◽  
Shallow Water Waves ◽  
Flow Equations ◽  
Hydraulic Flow ◽  
New Class ◽  
Simple Waves ◽  
The Waves

This paper describes the behaviour of large amplitude, long gravity waves as they move over a horizontal bed into a region where the flow is steady but sheared in a vertical direction. A new class of exact solutions to the nonlinear hydraulic flow equations is derived. These solutions describe progressing waves and are sufficiently general to allow both the shape of the free surface at any instant and the shear profile of the undisturbed flow to be specified. The waves are examples of neutrally stable disturbances in the sense that they neither grow nor decay in amplitude, although, like simple waves on an unsheared flow, they can break.

Surface Gravity Waves

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Water Surface ◽  
Wave Pattern ◽  
The Body ◽  
Surface Gravity ◽  
Shallow Water Waves ◽  
Ship Waves ◽  
Surface Gravity Waves ◽  
Basic Fluid

This chapter deals with the underlying mathematics of surface gravity waves, defined as gravity waves observed on an air–sea interface of the ocean. Surface gravity waves, or surface waves, differ from internal waves, gravity waves that occur within the body of the water (such as between parts of different densities). Examples of gravity waves are wind-generated waves on the water surface, as well tsunamis and ocean tides. Wind-generated gravity waves on the free surface of the Earth's seas, oceans, ponds, and lakes have a period of between 0.3 and 30 seconds. The chapter first describes the basic fluid equations before discussing the dispersion relations, with a particular focus on deep water waves, shallow water waves, and wavepackets. It also considers ship waves and how dispersion affects the wave pattern produced by a moving object, along with long and short waves.

scholarly journals Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity

2015 ◽  
Vol 145 (4) ◽  
pp. 791-883 ◽  
Author(s):  
M. D. Groves ◽  
E. Wahlén
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Applied Mathematics ◽  
Finite Depth ◽  
Constant Vorticity ◽  
Existence And Stability ◽  
Strong Surface ◽  
Weak Surface ◽  
The Stability ◽  
The Waves

We present an existence and stability theory for gravity–capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy𝓗subject to the constraint𝓘= 2µ, where𝓘is the wave momentum and 0 <µ≪ 1. Since𝓗and𝓘are both conserved quantities, a standard argument asserts the stability of the setDµof minimizers: solutions starting nearDµremain close toDµin a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg–de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation asµ↓ 0.

Refraction of gravity waves by weak current gradients

2001 ◽  
Vol 442 ◽  
pp. 157-159 ◽  
Author(s):  
KRISTIAN B. DYSTHE
Keyword(s):  
Surface Waves ◽  
Group Velocity ◽  
Gravity Waves ◽  
Deep Water ◽  
Current Velocity ◽  
Water Surface ◽  
Simple Formula ◽  
Vertical Component ◽  
Weak Current ◽  
The Waves

When deep water surface waves cross an area with variable current, refraction takes place. If the group velocity of the waves is much larger than the current velocity we show that the curvature of a ray, χ, is given by the simple formula χ = ζ/vg. Here ζ is the vertical component of the current vorticity and vg is the group velocity.

Effect of vorticity on steady water waves

2008 ◽  
Vol 608 ◽  
pp. 197-215 ◽  
Author(s):  
JOY KO ◽  
WALTER STRAUSS
Keyword(s):  
Shear Layer ◽  
Stagnation Point ◽  
Mass Flux ◽  
Water Waves ◽  
Large Amplitude ◽  
Maximum Amplitude ◽  
Finite Depth ◽  
Hydraulic Head ◽  
Two Dimensional

Two-dimensional, finite-depth periodic steady water waves with variable vorticity ω=γ(ψ) and large amplitude a are computed for a large number of cases. In particular, the effect of a shear layer at the top, the middle or the bottom is considered. The maximum amplitude amax varies monotonically with the vorticity function γ(ċ). It is increasing if the stagnation point is at the crest, and is decreasing if the stagnation point is in the interior of the fluid or on the bottom. Relationships between the amplitude, hydraulic head, depth and mass flux are investigated.

scholarly journals Flow structure beneath rotational water waves with stagnation points

2017 ◽  
Vol 812 ◽  
pp. 792-814 ◽  
Author(s):  
Roberto Ribeiro ◽  
Paul A. Milewski ◽  
André Nachbin
Keyword(s):  
Water Waves ◽  
Wave Speed ◽  
Physical Domain ◽  
Periodic Surface ◽  
Stagnation Points ◽  
Constant Vorticity ◽  
Interior Flow ◽  
The Waves ◽  
Particle Paths ◽  
Irrotational Wave

The purpose of this work is to explore in detail the structure of the interior flow generated by periodic surface waves on a fluid with constant vorticity. The problem is mapped conformally to a strip and solved numerically using spectral methods. Once the solution is known, the streamlines, pressure and particle paths can be found and mapped back to the physical domain. We find that the flow beneath the waves contains zero, one, two or three stagnation points in a frame moving with the wave speed, and describe the bifurcations between these flows. When the vorticity is sufficiently strong, the pressure in the flow and on the bottom boundary also has very different features from the usual irrotational wave case.

scholarly journals Nonlinear low-frequency gravity waves in a water-filled cylindrical vessel subjected to high-frequency excitations

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120536 ◽  
Author(s):  
Chunyan Zhou ◽  
Dajun Wang ◽  
Song Shen ◽  
Jing Tang Xing
Keyword(s):  
Gravity Waves ◽  
High Frequency ◽  
Water Waves ◽  
Large Amplitude ◽  
Low Frequency ◽  
Approximation Model ◽  
Fluid Structure Interactions ◽  
Governing Equations ◽  
Low Frequencies ◽  
Nonlinear Fluid

In the experiments of a water storage cylindrical shell, excited by a horizontal external force of sufficient large amplitude and high frequency, it has been observed that gravity water waves of low frequencies may be generated. This paper intends to investigate this phenomenon in order to reveal its mechanism. Considering nonlinear fluid–structure interactions, we derive the governing equations and the numerical equations describing the dynamics of the system, using a variational principle. Following the developed generalized equations, a four-mode approximation model is proposed with which an experimental case example is studied. Numerical calculation and spectrum analysis demonstrate that an external excitation with sufficient large amplitude and high frequency can produce gravity water waves with lower frequencies. The excitation magnitude and frequencies required for onset of the gravity waves are found based on the model. Transitions between different gravity waves are also revealed through the numerical analysis. The findings developed by this method are validated by available experimental observations.

scholarly journals THE SELF-MODULATION OF STRONGLY NONLINEAR WATER WAVES. INCOMPLETE RECURRENCE

2019 ◽  
Vol 47 (1) ◽  
pp. 116-117
Author(s):  
A.V. Slunyaev ◽  
A.S. Dosaev
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Euler Equations ◽  
Gravity Waves ◽  
Water Waves ◽  
Water Surface ◽  
The Self ◽  
Hydrodynamic Equations ◽  
Nonlinear Water Waves ◽  
Strongly Nonlinear

The processes of spontaneous self-modulation of steep gravity waves on the water surface with the formation of very short groups are investigated by means of the numerical simulation of the primitive Euler equations. It is shown that the subsequent demodulation is incomplete as a result of the generation of new waves with other lengths propagating both along the way and towards the main wave. Thus, in the framework of the full hydrodynamic equations an approximate analogue corresponds to the breather solution of the nonlinear Schrödinger equation. The parts of the research was supported by the RSF grant No 16-17-00041 and by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

Some effects of surface tension on steep water waves. Part 2

1980 ◽  
Vol 96 (3) ◽  
pp. 417-445 ◽  
Author(s):  
S. J. Hogan
Keyword(s):  
Surface Tension ◽  
Gravity Waves ◽  
Water Waves ◽  
Expansion Method ◽  
Capillary Waves ◽  
Maximum Wave Height ◽  
Pure Gravity ◽  
Wave Profiles ◽  
The Waves ◽  
Wave Properties

This paper continues an investigation of the effects of surface tension on steep water waves in deep water begun in Hogan (1979a). A Stokes-type expansion method is given which can be applied to most wavelengths. For capillary waves (2 cm or less) it is found that the surface of the highest wave encloses a bubble of air, as was found for pure capillary waves by Crapper (1957). For intermediate waves (20 cm) the wave profiles are similar to those of pure gravity waves and the wave properties increase monotonically. For gravity waves (200 cm) the wave properties all exhibit a maximum just short of the maximum wave height obtained by the method. The integral properties for all the waves are drawn and given in numerical form in the appendix.

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