An experimental investigation of breaking waves produced by a towed hydrofoil

1981 ◽  
Vol 377 (1770) ◽  
pp. 331-348 ◽  
Keyword(s):  
Surface Profile ◽  
Phase Speed ◽  
Breaking Waves ◽  
Finite Amplitude ◽  
Stokes Wave ◽  
Two Dimensional ◽  
Vertical Thickness ◽  
Turbulent Momentum ◽  
The Vertical Distribution ◽  
Turbulent Water

This paper presents the results of experiments on breaking waves produced by towing a submerged, two-dimensional hydrofoil at constant depth and speed. The wave field consists of a breaker followed by a train of lower, non-breaking waves. The breaker has a small zone of turbulent water riding its forward slope; this zone is called the breaking region. Measurements were made of surface height profiles, the vertical distribution of mean horizontal velocity in the wake of the wave, and the vertical thickness of the wake. The results support the hypothesis that the breaking region imparts a shearing force along the forward slope equal to the component of its weight in that direction. The force produces a turbulent, momentum-deficient wake similar to the wake of a towed, two-dimensional body in an infinite fluid. The vertical thickness of the wake grows in proportion to the square root of distance behind the breaker. The momentum deficit is approximately equal to the maximum momentum flux of a Stokes wave with the same phase speed as the breaker. The surface profile measurements yield several results: the proper independent variables describing the wave are its speed and the slope of its forward face. The relation between breaking wavelength and speed follows the finite-amplitude Stokes wave equation. The amplitude and the vertical extent of the breaking region are both proportional to the phase speed squared; however, they are not functions of the slope of the forward face of the wave. The breaking region has a small oscillation in its length with a regular period of 4.4 the period of a wave with phase speed equal to the hydrofoil speed. The amplitude of the oscillation diminishes with time. It is believed that this oscillation is due to wave components produced when the foil is started from rest.

Lagrangian moments and mass transport in Stokes waves

1987 ◽  
Vol 179 ◽  
pp. 547-555 ◽  
Author(s):  
Michael Longuet-Higgins
Keyword(s):  
Mass Transport ◽  
Phase Speed ◽  
Simple Relation ◽  
Finite Amplitude ◽  
Stokes Wave ◽  
Random Waves ◽  
The Third ◽  
Stokes Waves ◽  
Mass Transport Velocity ◽  
High Degree

The orbital motions in surface gravity waves are of interest for analysing wave records made by accelerometer buoys. In this paper we derive some exact expressions for the first, second and third cumulants of the vertical orbital displacements in a regular Stokes wave of finite amplitude in terms of previously known integral quantities of the wave: the kinetic and potential energies, the phase speed c and the mass-transport velocity U at the free surface. These results generalize a remarkably simple relation found previously between the Lagrangian-mean surface level and the product Uc.Expansions are given in powers of the wave steepness parameter ak which show that the third Lagrangian cumulant is very small – of order (ak)6, indicating a high degree of vertical symmetry in the orbit. This contrasts with the situation in random waves, where the third cumulant is of order (ak)4 only. It is shown that the increased skewness in random waves is due mainly to an O(ak)2 shift in the Lagrangian mean level of individual waves. Such shifts in mean level may be too gradual to be fully detected by some accelerometer buoys. In that case the apparent skewness will be reduced.

Potential energy in steep and breaking waves

1994 ◽  
Vol 278 ◽  
pp. 201-228 ◽  
Author(s):  
William W. Schultz ◽  
Jin Huh ◽  
Owen M. Griffin
Keyword(s):  
Potential Energy ◽  
Wave Height ◽  
Energy Input ◽  
Breaking Waves ◽  
Smooth Transition ◽  
Stokes Wave ◽  
Two Dimensional ◽  
Spectral Algorithm ◽  
Deep Water Wave ◽  
Experimental Scatter

We find that the RMS wave height (square root of the potential energy) rather than peak-to-peak wave height is a better experimental and analytic criterion for determining when a regular, two-dimensional deep-water wave will break. A spectral algorithm for two-dimensional potential flow is developed and used to compare breaking onset criteria for energy input from (i) converging sidewalls, (ii) a submerged disturbance, and (iii) wave focusing. We also find that wave-breaking criteria (potential energy or the more classical peak-to-peak wave height) are a function of the rate of energy input. Large plunging waves occur when energy input rates are large. As energy input rates become smaller there is a smooth transition to smaller spilling waves. The various energy input methods show similar breaking trends in the limit as the energy input rate becomes small - waves break when the potential energy becomes approximately 52 % of the energy for the most energetic Stokes wave, with the formation of a singularity immediately before the crest. The effects of wave modulation and reflection are briefly discussed and shown not to affect the potential energy breaking criterion significantly. The experimental scatter of the RMS wave height is shown to be half that of wave steepness during incipient breaking in wave packets.

A Note on the Evaluation of the Wave Resistance of Two-Dimensional Bodies from Measurements of the Downstream Wave Profile

1983 ◽  
Vol 27 (02) ◽  
pp. 90-92
Author(s):  
James H. Duncan
Keyword(s):  
Wave Train ◽  
Wave Theory ◽  
Stokes Wave ◽  
Two Dimensional ◽  
Third Order ◽  
The Subject ◽  
Lord Kelvin ◽  
The Waves ◽  
The Vertical Distribution

As a body moves horizontally at constant speed in the proximity of a free surface it experiences a resistance due to the generation of waves. In two-dimensional cases the determination of this resistance from properties of the wave train has been the subject of several investigations. The linear theory was first presented by Lord Kelvin [1] 2 and later by Havelock [2] and Lamb [8]. Wehausen and Laitone [4] have derived an exact resistance formula in terms of the vertical distribution of velocity in the waves and the downstream surface height profile. This formula was later evaluated by Salvesen and von Kerczek [5] using third-order Stokes wave theory.

Computations of fully nonlinear three-dimensional wave–wave and wave–body interactions. Part 1. Dynamics of steep three-dimensional waves

2001 ◽  
Vol 438 ◽  
pp. 11-39 ◽  
Author(s):  
MING XUE ◽  
HONGBO XÜ ◽  
YUMING LIU ◽  
DICK K. P. YUE
Keyword(s):  
Three Dimensional ◽  
Breaking Waves ◽  
Stokes Wave ◽  
Two Dimensional ◽  
Wave Instability ◽  
Fully Nonlinear ◽  
Stokes Waves ◽  
Distinct Features ◽  
Wave Problems ◽  
Dimensional Wave

We develop an efficient high-order boundary-element method with the mixed-Eulerian–Lagrangian approach for the simulation of fully nonlinear three-dimensional wave–wave and wave–body interactions. For illustration, we apply this method to the study of two three-dimensional steep wave problems. (The application to wave–body interactions is addressed in an accompanying paper: Liu, Xue & Yue 2001.) In the first problem, we investigate the dynamics of three-dimensional overturning breaking waves. We obtain detailed kinematics and full quantification of three-dimensional effects upon wave plunging. Systematic simulations show that, compared to two-dimensional waves, three-dimensional waves generally break at higher surface elevations and greater maximum longitudinal accelerations, but with smaller tip velocities and less arched front faces. For the second problem, we study the generation mechanism of steep crescent waves. We show that the development of such waves is a result of three-dimensional (class II) Stokes wave instability. Starting with two-dimensional Stokes waves with small three-dimensional perturbations, we obtain direct simulations of the evolution of both L2 and L3 crescent waves. Our results compare quantitatively well with experimental measurements for all the distinct features and geometric properties of such waves.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

A Two-Dimensional Surface Profile Imaging Technique Based on Heterodyne Interferometer

2005 ◽  
Vol 295-296 ◽  
pp. 477-482
Author(s):  
K.W. Wang ◽  
Z.J. Cai ◽  
Li Jiang Zeng
Keyword(s):  
High Speed ◽  
Imaging Technique ◽  
Surface Profile ◽  
Ccd Camera ◽  
Measurement Range ◽  
Interference Signal ◽  
Two Dimensional ◽  
Step Method ◽  
Heterodyne Interferometer ◽  
Dimensional Surface

A two-dimensional surface profile imaging technique based on heterodyne interferometer is proposed. A piezo translator vibrated grating is used to generate a heterodyne signal. A high speed CCD camera is used to extract the interference signal using a five step method. The uncertainty in the displacement measurement is approximately 0.035 µm within a measurement range of 1.7 µm, confirming the two dimensional heterodyne interferometer is valid for measuring the surface profile. The method is also available for low coherence heterodyne interferometer due to the optical frequency shifts caused by the vibration of grating independent on the wavelength.

The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 471-488 ◽  
Keyword(s):  
Stream Function ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Travelling Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Finite Amplitude ◽  
Fundamental Wave ◽  
Wave Steepness ◽  
Horizontal Scale

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.

An experimental investigation of incipient spilling breakers

2009 ◽  
Vol 633 ◽  
pp. 271-283 ◽  
Author(s):  
J. D. DIORIO ◽  
X. LIU ◽  
J. H. DUNCAN
Keyword(s):  
Wave Breaking ◽  
High Speed ◽  
Phase Speed ◽  
Breaking Waves ◽  
Front Face ◽  
Geometrical Parameters ◽  
Instability Mechanism ◽  
Scaling Relationships ◽  
Self Similar ◽  
Spilling Breakers

In the present paper, the profiles of incipient spilling breaking waves with wavelengths ranging from 10 to 120cm were studied experimentally in clean water. Short-wavelength breakers were generated by wind, while longer-wavelength breakers were generated by a mechanical wavemaker, using either a dispersive focusing or a sideband instability mechanism. The crest profiles of these waves were measured with a high-speed cinematic laser-induced fluorescence technique. For all the wave conditions reported herein, wave breaking was initiated with a capillary-ripple pattern as described in Duncan et al. (J. Fluid Mech., vol. 379, 1999, pp. 191–222). In the present paper, it is shown that at incipient breaking the crest shape is self-similar with two geometrical parameters that depend only on the slope of a particular point on the front face of the gravity wave. The scaling relationships appear to be universal for the range of wavelengths studied herein and hold for waves generated by mechanical wavemakers and by wind. The slope measure is found to be dependent on the wave phase speed and the rate of growth of the crest height prior to incipient breaking.

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