A Modified Linear Lagrangian Model for Irregular Long-Crested Waves

2009 ◽  
Author(s):  
Se´bastien Fouques ◽  
Carl Trygve Stansberg
Keyword(s):  
Mass Conservation ◽  
Second Order ◽  
Lagrangian Model ◽  
Wave Crest ◽  
Lagrangian Description ◽  
Wave Steepness ◽  
Linear Solution ◽  
Fluid Particles ◽  
Wave Properties ◽  
Crest Height

Wave crest height and steepness are crucial parameters for the design of ships and offshore structures. For irregular sea states, they are commonly predicted by using linear wave theory and a Eulerian description of the fluid motion. This theory only applies when the wave steepness is small and it fails to capture extreme wave events. Such linear solutions can also be extended by including second-order terms in order to provide more realistic wave properties. The paper describes a model for irregular long-crested waves that is based on a modified linear solution derived from a Lagrangian description of the fluid, i.e. by considering the motion of individual fluid particles. Lagrangian solutions have the advantage of showing realistic wave profiles with sharp crests and broad troughs already at the first order, whereas these features only appear at the second order when using the Eulerian approach. Still, a severe drawback with the former is that the mass conservation is not fulfilled exactly. The aim of the modification in the present Lagrangian model is to ensure that the mass conservation is always fulfilled in the solution. This is done by using the second-order residual in the continuity equation to lift up the fluid particles vertically. Comparative investigations of wave properties such as the crest height and the wave steepness are further carried out by making use of both numerical case studies and wave tank recordings. The wave models used in the comparisons include linear and second-order Eulerian solutions as well as the modified linear Lagrangian one.

A Second Order Lagrangian Model for Irregular Ocean Waves

2005 ◽  
Vol 128 (3) ◽  
pp. 177-183 ◽  
Author(s):  
Sébastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

New Insights in Extreme Crest Height Distributions: A Summary of the ‘CresT’ JIP

2011 ◽  
Author(s):  
Bas Buchner ◽  
George Forristall ◽  
Kevin Ewans ◽  
Marios Christou ◽  
Janou Hennig
Keyword(s):  
Single Point ◽  
Field Measurements ◽  
Order Theory ◽  
Second Order ◽  
Wave Crest ◽  
Height Distribution ◽  
Extreme Waves ◽  
Wave Height Distribution ◽  
Floating Platforms ◽  
Crest Height

The objective of the CresT JIP was ‘to develop models for realistic extreme waves and a design methodology for the loading and response of floating platforms’. Within this objective the central question was: ‘What is the highest (most critical) wave crest that will be encountered by my platform in its lifetime?’ Based on the presented results for long and short-crested numerical, field and basin results in the paper, it can be concluded that the statistics of long-crested waves are different than those of short-crested waves. But also short-crested waves show a trend to reach crest heights above second order. This is in line with visual observations of the physics involved: crests are sharper than predicted by second order, waves are asymmetric (fronts are steeper) and waves are breaking. Although the development of extreme waves within short-crested sea states still needs further investigation (including the counteracting effect of breaking), at the end of the CresT project the following procedure for taking into account extreme waves in platform design is recommended: 1. For the wave height distribution, use the Forristall distribution (Forristall, 1978). 2. For the crest height distribution, use 2nd order distribution as basis. 3. Both the basin and field measurements show crest heights higher than predicted by second order theory for steeper sea states. It is therefore recommended to apply a correction to the second order distribution based on the basin results. 4. Account for the sampling variability at the tail of the distribution (and resulting remaining possibility of higher crests than given by the corrected second order distribution) in the reliability analysis. 5. Consider the fact that the maximum crest height under a complete platform deck can be considerably higher than the maximum crest at a single point.

A Second Order Lagrangian Model for Irregular Ocean Waves

2004 ◽  
Author(s):  
Se´bastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic Aperture Radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, the slope and the mean curvature are studied.

Analysis and Applications of Second Order Models for Maximum Crest Height

2002 ◽  
Author(s):  
Harald E. Krogstad ◽  
Stephen F. Barstow
Keyword(s):  
Time Series ◽  
Wave Height ◽  
Water Depth ◽  
Second Order ◽  
Wave Steepness ◽  
Data Set ◽  
Wave Data ◽  
Definite Improvement ◽  
Crest Height

Expressions for the maximum crest height are reviewed and tested on data from five different sensors in the WACSIS data set. The overall agreement is good and the analysis supports that second order models give accurate expressions for the distribution of the maximum crest height for varying water depth and wave steepness. In the second part of the paper, the expressions are combined with the existing extreme crest and wave height framework and applied to sets of time series and long term wave data. It is concluded that the 2nd order models represent a definite improvement over earlier empirical parametrizations.

Analysis and Applications of Second-Order Models for Maximum Crest Height

2004 ◽  
Vol 126 (1) ◽  
pp. 66-71 ◽  
Author(s):  
Harald E. Krogstad ◽  
Stephen F. Barstow
Keyword(s):  
Time Series ◽  
Wave Height ◽  
Water Depth ◽  
Second Order ◽  
Wave Steepness ◽  
Data Set ◽  
Wave Data ◽  
Definite Improvement ◽  
Crest Height

Expressions for the maximum crest height are reviewed and tested on data from five different sensors in the WACSIS data set. The overall agreement is good and the analysis supports that second-order models give accurate expressions for the distribution of the maximum crest height for varying water depth and wave steepness. In the second part of the paper, the expressions are combined with the existing extreme crest and wave height framework and applied to sets of time series and long term wave data. It is concluded that the second-order models represent a definite improvement over earlier empirical parametrizations.

Second Order Model for Wave Crests Used in Prediction of Green Water Load and Volume on Ships in Random Waves

2005 ◽  
Vol 128 (4) ◽  
pp. 271-275 ◽  
Author(s):  
Hanne Therese Wist ◽  
Dag Myrhaug ◽  
Håvard Rue
Keyword(s):  
Prediction Method ◽  
Second Order ◽  
Green Water ◽  
Wave Crest ◽  
Water Load ◽  
Load Line ◽  
Long Term Prediction ◽  
Height Model ◽  
Crest Height

The probability that a wave crest in a random sea will exceed a specified height has long been recognized as important statistics in practical work, e.g., in predicting green water load and volume on a ship. Nonlinear probability density functions for predicting green water load and volume are presented. The models are based on the parametric model of Ogawa (2003, “Long-Term Prediction Method for the Green Water Load and Volume for an Assessment of the Load Line,” J. Marine Sci. Technol., 7, pp. 137–144) combined with transformation of a second order wave crest height model. The wave crest height model is obtained from second order wave theory for a narrow-banded sea state in combination with transformation of the Rayleigh distribution. Results from the second order models are compared with model tests of a cargo ship presented in Ogawa (2003, “Long-Term Prediction Method for the Green Water Load and Volume for an Assessment of the Load Line,” J. Marine Sci. Technol., 7, pp. 137–144) and the Ogawa models.

Wave Crest Sensor Intercomparison Study: An Overview of WACSIS

2002 ◽  
Author(s):  
George Z. Forristall ◽  
Stephen F. Barstow ◽  
Harald E. Krogstad ◽  
Marc Prevosto ◽  
Paul H. Taylor ◽  
...  
Keyword(s):  
Second Order ◽  
Wave Crest ◽  
Height Distribution ◽  
High Frequencies ◽  
Video Cameras ◽  
Period Distribution ◽  
Wave Sensors ◽  
Intercomparison Study ◽  
Crest Height ◽  
Made In

The Wave Crest Sensor Intercomparison Study (WACSIS) was designed as a thorough investigation of the statistical distribution of crest heights. Measurements were made in the southern North Sea during the winter of 1997–1998 from the Meetpost Noordwijk in 18 m water depth. The platform was outfitted with several popular wave sensors, including Saab and Marex radars, an EMI laser, a Baylor wave staff and a Vlissingen step gauge. Buoys were moored nearby to obtain directional spectra. Two video cameras viewed the ocean under the wave sensors and their signals were recorded digitally. The data analysis focused on comparisons of the crest height measurements from the various sensors and comparisons of the crest height distributions derived from the sensors and from theories. Some of the sensors had greater than expected energy at high frequencies. Once the measurements were filtered at 0.64 Hz, the Baylor, EMI and Vlissingen crest height distributions matched quite closely, while those from the other sensors were a few percent higher. The Baylor and EMI crest distributions agreed very well with the statistics from second order simulations, while previous parameterizations of the crest height distribution were generally too high. We conclude that crest height distributions derived from second order simulations can be used with confidence in engineering calculations. The data were also used in investigations of crest and trough shapes and the joint height/period distribution.

Second Order Model for Wave Crests Used in Prediction of Green Water Load and Volume on Ships in Random Waves

2004 ◽  
Author(s):  
Hanne Therese Wist ◽  
Dag Myrhaug ◽  
Ha˚vard Rue
Keyword(s):  
Wave Theory ◽  
Rayleigh Distribution ◽  
Second Order ◽  
Green Water ◽  
Wave Crest ◽  
Random Waves ◽  
Sea State ◽  
Water Load ◽  
Height Model ◽  
Crest Height

The probability that a wave crest in a random sea will exceed a specified height has long been recognized as important statistics in practical work, e.g., in predicting green water load and volume on a ship. Nonlinear probability density functions for predicting green water load and volume are presented. The models are based on the linear model of [1] in combination with transformation of a second order wave crest height model. The wave crest height model is obtained from second order wave theory for a narrow-banded sea state in combination with transformation of the Rayleigh distribution. Results from the models are compared with model tests of a cargo ship presented in [1].

Wave Crest Sensor Intercomparison Study: An Overview of WACSIS

2004 ◽  
Vol 126 (1) ◽  
pp. 26-34 ◽  
Author(s):  
George Z. Forristall ◽  
Stephen F. Barstow ◽  
Harald E. Krogstad ◽  
Marc Prevosto ◽  
Paul H. Taylor ◽  
...  
Keyword(s):  
Second Order ◽  
Wave Crest ◽  
Height Distribution ◽  
High Frequencies ◽  
Video Cameras ◽  
Period Distribution ◽  
Wave Sensors ◽  
Intercomparison Study ◽  
Crest Height ◽  
Made In

The Wave Crest Sensor Intercomparison Study (WACSIS) was designed as a thorough investigation of the statistical distribution of crest heights. Measurements were made in the southern North Sea during the winter of 1997–1998 from the Meetpost Noordwijk in 18 m water depth. The platform was outfitted with several popular wave sensors, including Saab and Marex radars, an EMI laser, a Baylor wave staff and a Vlissingen step gauge. Buoys were moored nearby to obtain directional spectra. Two video cameras viewed the ocean under the wave sensors and their signals were recorded digitally. The data analysis focused on comparisons of the crest height measurements from the various sensors and comparisons of the crest height distributions derived from the sensors and from theories. Some of the sensors had greater than expected energy at high frequencies. Once the measurements were filtered at 0.64 Hz, the Baylor, EMI and Vlissingen crest height distributions matched quite closely, while those from the other sensors were a few percent higher. The Baylor and EMI crest distributions agreed very well with the statistics from second order simulations, while previous parameterizations of the crest height distribution were generally too high. We conclude that crest height distributions derived from second order simulations can be used with confidence in engineering calculations. The data were also used in investigations of crest and trough shapes and the joint height/period distribution.

Non-Gaussian Extreme Waves in the Central North Sea

1997 ◽  
Vol 119 (3) ◽  
pp. 146-150 ◽  
Author(s):  
J. Skourup ◽  
N.-E. O. Hansen ◽  
K. K. Andreasen
Keyword(s):  
Wave Height ◽  
North Sea ◽  
Upper Bound ◽  
Wave Crest ◽  
Extreme Waves ◽  
Freak Waves ◽  
Freak Wave ◽  
Wave Trains ◽  
Central North Sea ◽  
Crest Height

The area of the Central North Sea is notorious for the occurrence of very high waves in certain wave trains. The short-term distribution of these wave trains includes waves which are far steeper than predicted by the Rayleigh distribution. Such waves are often termed “extreme waves” or “freak waves.” An analysis of the extreme statistical properties of these waves has been made. The analysis is based on more than 12 yr of wave records from the Mærsk Olie og Gas AS operated Gorm Field which is located in the Danish sector of the Central North Sea. From the wave recordings more than 400 freak wave candidates were found. The ratio between the extreme crest height and the significant wave height (20-min value) has been found to be about 1.8, and the ratio between extreme crest height and extreme wave height has been found to be 0.69. The latter ratio is clearly outside the range of Gaussian waves, and it is higher than the maximum value for steep nonlinear long-crested waves, thus indicating that freak waves are not of a permanent form, and probably of short-crested nature. The extreme statistical distribution is represented by a Weibull distribution with an upper bound, where the upper bound is the value for a depth-limited breaking wave. Based on the measured data, a procedure for determining the freak wave crest height with a given return period is proposed. A sensitivity analysis of the extreme value of the crest height is also made.

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