The Set-Down and Set-Up of Directionally Spread and Crossing Surface Gravity Wave Groups in Severe North Sea Storms

2018 ◽  
Author(s):  
Mark L. McAllister ◽  
Thomas A. A. Adcock ◽  
Ton S. van den Bremer ◽  
Paul H. Taylor
Keyword(s):  
Free Surface ◽  
North Sea ◽  
Water Depth ◽  
Order Theory ◽  
Relative Magnitude ◽  
Second Order ◽  
Experimental Results ◽  
Surface Gravity Wave ◽  
Wave Groups ◽  
Set Up

Recent work by McAllister et al. (2018) [1] has experimentally confirmed that the set-down of the wave-averaged free surface, first described by Longuet-Higgins and Stewart (1962) [2], can turn into a set-up when wave groups are sufficiently spread or cross at large angles. Experimental results were shown to agree well with second-order theory, including frequency-sum and frequency-difference terms, where the latter are responsible for the wave-averaged free surface. In this paper, we review these experimental results and examine theoretically the magnitude of the wave-averaged free surface in realistic extreme North Sea conditions. Specifically, we examine the role of the shape of the spectrum, water depth, and the relative magnitude of the peak frequencies of the two crossing groups. We find that having a realistic spectrum (JONSWAP vs. Gaussian) considerably enhances the magnitude of the second-order contribution, the total second-order signal increases with decreasing depth and can display a maximum provided the water depth is shallow enough for small to moderate degrees of spreading or crossing angles and is larger for spectral peaks that are further apart.

scholarly journals SHOALING PROPERTIES OF BOUNDED LONG WAVES

1984 ◽  
Vol 1 (19) ◽  
pp. 54 ◽  
Author(s):  
E.P.D. Mansard ◽  
V. Barthel
Keyword(s):  
Shallow Water ◽  
Water Depth ◽  
Second Order ◽  
Long Waves ◽  
Experimental Results ◽  
Wave Groups ◽  
Good Insight ◽  
Set Up ◽  
Theoretical Considerations ◽  
Insight Into

Group bounded long waves which appear as a set-down under a group of high waves and a set-up in between groups are well described for constant water depth. However, their propagation into shallow water and their interaction with the constituent wave groups are not well understood and theoretically described yet. Therefore, model investigations were carried out to study shoaling properties of these second order waves in terms of amplitudes and phases. The tests give a good insight into the phenomenon and suggest distinct shoaling properties. Moreover, experimental results provide a valuable basis for future theoretical considerations.

scholarly journals The set-down and set-up of directionally spread and crossing surface gravity wave groups

2017 ◽  
Vol 835 ◽  
pp. 131-169 ◽  
Author(s):  
M. L. McAllister ◽  
T. A. A. Adcock ◽  
P. H. Taylor ◽  
T. S. van den Bremer
Keyword(s):  
Free Surface ◽  
Gravity Wave ◽  
Standing Wave ◽  
Wave Pattern ◽  
Surface Gravity ◽  
Linear Wave ◽  
Surface Gravity Wave ◽  
Wave Groups ◽  
Single Wave ◽  
Set Up

For sufficiently directionally spread surface gravity wave groups, the set-down of the wave-averaged free surface, first described by Longuet-Higgins and Stewart (J. Fluid Mech. vol. 13, 1962, pp. 481–504), can turn into a set-up. Using a multiple-scale expansion for two crossing wave groups, we examine the structure and magnitude of this wave-averaged set-up, which is part of a crossing wave pattern that behaves as a modulated partial standing wave: in space, it consists of a rapidly varying standing-wave pattern slowly modulated by the product of the envelopes of the two groups; in time, it grows and decays on the slow time scale associated with the translation of the groups. Whether this crossing wave pattern actually enhances the surface elevation at the point of focus depends on the phases of the linear wave groups, unlike the set-down, which is always negative and inherits the spatial structure of the underlying envelope(s). We present detailed laboratory measurements of the wave-averaged free surface, examining both single wave groups, varying the degree of spreading from small to very large, and the interaction between two wave groups, varying both the degree of spreading and the crossing angle between the groups. In both cases, we find good agreement between the experiments, our simple expressions for the set-down and set-up, and existing second-order theory based on the component-by-component interaction of individual waves with different frequencies and directions. We predict and observe a set-up for wave groups with a Gaussian angular amplitude distribution with standard deviations of above $30{-}40^{\circ }$ ($21{-}28^{\circ }$ for energy spectra), which is relatively large for realistic sea states, and for crossing sea states with angles of separation of $50{-}70^{\circ }$ and above, which are known to occur in the ocean.

scholarly journals Experimental particle paths and drift velocity in steep waves at finite water depth

2016 ◽  
Vol 810 ◽  
Author(s):  
John Grue ◽  
Jostein Kolaas
Keyword(s):  
Drift Velocity ◽  
Water Depth ◽  
Order Theory ◽  
Second Order ◽  
Velocity Shear ◽  
Vertical Position ◽  
Functional Relationships ◽  
Finite Water Depth ◽  
Particle Paths ◽  
Experimental Particle

The Lagrangian paths, horizontal Lagrangian drift velocity, $U_{L}$, and the Lagrangian excess period, $T_{L}-T_{0}$, where $T_{L}$ is the Lagrangian period and $T_{0}$ the Eulerian linear period, are obtained by particle tracking velocimetry (PTV) in non-breaking periodic laboratory waves at a finite water depth of $h=0.2~\text{m}$, wave height of $H=0.49h$ and wavenumber of $k=0.785/h$. Both $U_{L}$ and $T_{L}-T_{0}$ are functions of the average vertical position of the paths, $\bar{Y}$, where $-1<\bar{Y}/h<0$. The functional relationships $U_{L}(\bar{Y})$ and $T_{L}-T_{0}=f(\bar{Y})$ are very similar. Comparisons to calculations by the inviscid strongly nonlinear Fenton method and the second-order theory show that the streaming velocities in the boundary layers below the wave surface and above the fluid bottom contribute to a strongly enhanced forward drift velocity and excess period. The experimental drift velocity shear becomes more than twice that obtained by the Fenton method, which again is approximately twice that of the second-order theory close to the surface. There is no mass flux of the periodic experimental waves and no pressure gradient. The results from a total number of 80 000 experimental particle paths in the different phases and vertical positions of the waves show a strong collapse. The particle paths are closed at the two vertical positions where $U_{L}=0$.

scholarly journals Did the Draupner wave occur in a crossing sea?

2011 ◽  
Vol 467 (2134) ◽  
pp. 3004-3021 ◽  
Author(s):  
T. A. A. Adcock ◽  
P. H. Taylor ◽  
S. Yan ◽  
Q. W. Ma ◽  
P. A. E. M. Janssen
Keyword(s):  
Low Frequency ◽  
Quality Measurement ◽  
Order Theory ◽  
Second Order ◽  
Flow Solver ◽  
The North ◽  
Highly Nonlinear ◽  
Nonlinear Potential ◽  
Wind Sea ◽  
Set Up

The ‘New Year Wave’ was recorded at the Draupner platform in the North Sea and is a rare high-quality measurement of a ‘freak’ or ‘rogue’ wave. The wave has been the subject of much interest and numerous studies. Despite this, the event has still not been satisfactorily explained. One piece of information that was not directly measured at the platform, but which is vital to understanding the nonlinear dynamics is the wave's directional spreading. This paper investigates the directionality of the Draupner wave and concludes it might have resulted from two wave-groups crossing, whose mean wave directions were separated by about 90 ° or more. This result has been deduced from a set-up of the low-frequency second-order difference waves under the giant wave, which can be explained only if two wave systems are propagating at such an angle. To check whether second-order theory is satisfactory for such a highly nonlinear event, we have run numerical simulations using a fully nonlinear potential flow solver, which confirm the conclusion deduced from the second-order theory. This is backed up by a hindcast from European Centre for Medium-Range Weather Forecasts that shows swell waves propagating at approximately 80 ° to the wind sea. Other evidence that supports our conclusion are the measured forces on the structure, the magnitude of the second-order sum waves and some other instances of freak waves occurring in crossing sea states.

Nonlinear Wave Forces Acting on Submerged Horizontal Cylinders

1988 ◽  
Vol 110 (1) ◽  
pp. 62-70 ◽  
Author(s):  
R. Inoue ◽  
Y. Kyozuka
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Nonlinear Wave ◽  
Wave Breaking ◽  
Nonlinear Effects ◽  
Second Order ◽  
Experimental Results ◽  
Wave Forces ◽  
Regular Perturbation ◽  
Horizontal Cylinders

This paper is to present experimental results of the first and second-order wave forces acting on three kinds of horizontally submerged cylinders. Wave height, wave frequency and the models’ submergence were varied in the experiments. These results are compared with the numerical calculations based on the regular perturbation theory. Through this study, it was found that the calculations of both the first and second-order wave forces coincide with the experiments when the cylinders are submerged at a sufficient depth. However, in the case that the cylinders are close to the free surface and/or wave amplitudes are relatively large, the experimental results become small compared with the calculations because of nonlinear effects, such as wave breaking observed in the experiments.

The Prediction of Aerofoil Pressure Distributions for Sub-Critical Viscous Flows

1970 ◽  
Vol 21 (3) ◽  
pp. 291-302 ◽  
Author(s):  
R. C. Lock ◽  
P. G. Wilby ◽  
B. J. Powell
Keyword(s):  
Boundary Layer ◽  
Approximate Method ◽  
Iterative Procedure ◽  
Viscous Flows ◽  
Order Theory ◽  
Second Order ◽  
Experimental Results ◽  
Inviscid Flows ◽  
Exact Theory ◽  
Pressure Distributions

SummaryThe paper first describes an approximate method for calculating inviscid flows round arbitrary aerofoils at sub-critical Mach numbers, based on second-order theory with empirical improvements to give better agreement with exact theory; several comparisons are shown. This method is then used as the basis of an iterative procedure for calculating the effect of the boundary layer on the surface pressures and overall forces; several comparisons are given with recent experimental results.

A Second-Order Theory for the Potential Flow About Thin Hydrofoils

1984 ◽  
Vol 28 (01) ◽  
pp. 55-64
Author(s):  
Colen Kennell ◽  
Allen Plotkin
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Potential Flow ◽  
Order Theory ◽  
Wave Drag ◽  
Second Order ◽  
The Body ◽  
Surface Shape ◽  
Surface Boundary ◽  
Free Surface Boundary Condition

This research addresses the potential flow about a thin two-dimensional hydrofoil moving with constant velocity at a fixed depth beneath a free surface. The thickness-to-chord ratio of the hydrofoil and disturbances to the free stream are assumed to be small. These small perturbation assumptions are used to produce first-and second-order subproblems structured to provide consistent approximations to boundary conditions on the body and the free surface. Nonlinear corrections to the free-surface boundary condition are included at second order. Each subproblem is solved by a distribution of sources and vortices on the chord line and doublets on the free surface. After analytic determination of source and doublet strengths, a singular integral equation for the vortex strength is derived. This integral equation is reduced to a Fredholm integral equation which is solved numerically. Lift, wave drag, and free-surface shape are calculated for a flat plate and a Joukowski hydrofoil. The importance of free-surface effects relative to body effects is examined by a parametric variation of Froude number and depth of submergence.

The Consistent Second-Order Wave Theory and Its Application to a Submerged Spheroid

1970 ◽  
Vol 14 (01) ◽  
pp. 23-50
Author(s):  
Young H. Chey
Keyword(s):  
Free Surface ◽  
Solid Wall ◽  
Three Dimensional ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Phenomena ◽  
First Order ◽  
Theoretical Results

Because of the recognized inadequacy of first-order linearized surface-wave theory, the author has developed, for a three-dimensional body, a new second-order theory which provides a better description of free-surface phenomena. The new theory more accurately satisfies the kinematic boundary condition on the solid wall, and takes into account the nonlinearity of the condition at the free surface. The author applies the new theory to a submerged spheroid, to calculate wave resistance. Experiments were conducted to verify the theory, and their results are compared with the theoretical results. The comparison indicates that the use of the new theory leads to more accurate prediction of wave resistance.

Nonlinear Crest Height Distribution in Three-Dimensional Ocean Waves

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
Felice Arena ◽  
Alfredo Ascanelli
Keyword(s):  
Water Depth ◽  
Three Dimensional ◽  
Ocean Waves ◽  
Order Theory ◽  
Second Order ◽  
Wave Crest ◽  
Height Distribution ◽  
Random Waves ◽  
Weakly Nonlinear ◽  
Wave Trough

The interest and studies on nonlinear waves are increased recently for their importance in the interaction with floating and fixed bodies. It is also well-known that nonlinearities influence wave crest and wave trough distributions, both deviating from the Rayleigh law. In this paper, a theoretical crest distribution is obtained, taking into account the extension of Boccotti’s quasideterminism theory (1982, “On Ocean Waves With High Crests,” Meccanica, 17, pp. 16–19), up to the second order for the case of three-dimensional waves in finite water depth. To this purpose, the Fedele and Arena (2005, “Weakly Nonlinear Statistics of High Random Waves,” Phys. Fluids, 17(026601), pp. 1–10) distribution is generalized to three-dimensional waves on an arbitrary water depth. The comparison with Forristall’s second order model (2000, “Wave Crest Distributions: Observations and Second-Order Theory,” J. Phys. Oceanogr., 30(8), pp. 1931–1943) shows the theoretical confirmation of his conclusion: The crest distribution in deep water for long-crested and short-crested waves are very close to each other; in shallow water the crest heights in three-dimensional waves are greater than values given by the long-crested model.

On higher-order wave theory for submerged two-dimensional bodies

1969 ◽  
Vol 38 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Nils Salvesen
Keyword(s):  
Free Surface ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Drag ◽  
Higher Order ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
Free Surface Waves ◽  
Body Shapes

The importance of non-linear free-surface effects on potential flow past two-dimensional submerged bodies is investigated by the use of higher-order perturbation theory. A consistent second-order solution for general body shapes is derived. A comparison between experimental data and theory is presented for the free-surface waves and for the wave resistance of a foil-shaped body. The agreement is good in general for the second-order theory, while the linear theory is shown to be inadequate for predicting the wave drag at the relatively small submergence treated here. It is also shown, by including the third-order freesurface effects, how the solution to the general wave theory breaks down at low speeds.

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