A note on finite depth second-order wave–wave interactions

Dalzell, J F

doi:10.1016/s0141-1187(99)00008-5

Cited by 137 publications

(132 citation statements)

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“…We compare detailed measurements conducted at the fully directional FloWave Ocean Energy Research Facility at the University of Edinburgh with existing multiple-component second-order wave theory (Sharma & Dean 1981;Dalzell 1999;Forristall 2000) and derive new results using a multiple-scale approach which lend greater insight into the problem. We believe that ours is the first experimental observation of set-up for highly directionally spread and crossing groups, following field observations of a set-up underneath the famous Draupner rogue wave (Walker et al 2004) and on Lake George, Australia (Toffoli et al 2007).…”

Section: Discussionmentioning

confidence: 99%

“…The coefficients δk and δθ correspond to the magnitude of the discrete steps, so that δk → dk as N k → ∞ and similarly for δθ . The corresponding second-order difference waves that represent the wave-averaged free surface may be calculated as (Dalzell 1999) figure 5 compares the component-by-component solution (2.21) (a-d) with the multiple-scale solution for the set-down (2.8) (e-h) for a single wave group, demonstrating good agreement for low degrees of spreading. The mean direction of wave group propagation is left to right in the positive x direction.…”

Section: Multicomponent Second-order Theory (Review)mentioning

confidence: 98%

“…By considering the linear signal as the sum of individual components with different frequencies travelling in different directions, Hasselmann (1962) implicitly and, much later yet explicitly, Sharma & Dean (1981), Dalzell (1999) and Forristall (2000) derived interaction kernels for the nonlinear bound harmonics at second order. We assume independence between the directional Ω and amplitudê η distributions, so that the linear signal is given by a summation over N k discrete components in N θ directions, 20) where the wavenumber vector k n,i = k n (cos(θ i ), sin(θ i )) has magnitude k n and θ is measured anticlockwise from the positive x-axis.…”

Section: Multicomponent Second-order Theory (Review)mentioning

confidence: 99%

“…The interaction kernels for the sum B + and difference B − terms at second order and for general water depth are given by (Dalzell 1999) Appendix B. Harmonic separation B.1. Two-phase harmonic extraction Two-phase harmonic extraction through the creation of crest-and trough-focus wave groups η 0 and η 180 allows for the separation of harmonics of odd and even powers in amplitude (Baldock et al 1996),…”

Section: Supplementary Moviesmentioning

confidence: 99%

“…Although expressions for the frequency-difference terms in a multidirectional sea can be distilled from Hasselmann (1962) (cf. Okihiro, Guza & Seymour 1992), Sharma & Dean (1981), Dalzell (1999) and Forristall (2000) are commonly credited for extending the work of Longuet-Higgins & Stewart (1962) to directional seas, allowing for interactions between parent wave components of different frequencies and travelling in different directions (see Pellet et al (2017) for a recent rederivation that also includes pressure). In the limit of a quasimonochromatic group with a single carrier wave travelling in one direction, differential equations describing these second-order bound interactions can also be calculated using a multiple-scale approach.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2017

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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• (21-28 • for energy spectra), which is relatively large for realistic sea states, and for crossing sea states with angles of separation of 50-70• and above, which are known to occur in the ocean.

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Section: Discussionmentioning

confidence: 99%

Section: Multicomponent Second-order Theory (Review)mentioning

confidence: 98%

Section: Multicomponent Second-order Theory (Review)mentioning

confidence: 99%

Section: Supplementary Moviesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2017

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Estimates of Lagrangian transport by surface gravity wave groups: The effects of finite depth and directionality

Bremer

Taylor

2015

JGR Oceans

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Two physical phenomena drive the Lagrangian trajectories of neutrally buoyant particles underneath surface gravity wave groups: the Stokes drift results in a net displacement of particles in the direction of propagation of the group, whereas the Eulerian return flow transports such particles in the opposite direction. Generally, the Stokes drift is the larger of the two near the surface, whereas the effects of the return flow dominate at depth. A transition depth can be defined that separates the two regimes. Using a multiple‐scales expansion, we provide leading‐order estimates of the forward transport, the backward transport, and the transition depth for realistic sea states. We consider the effects of both finite depth and the directionally spread nature of the waves on our estimates. We show that from the perspective of the return flow, almost all seas are of finite depth. In fact, many seas can be shown to be “shallow” from the perspective of the return flow with little variation of this flow with depth. Furthermore, even small degrees of directional spreading can considerably reduce the magnitude of the return flow and its transport.

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Non‐linear time‐domain models for irregular wave diffraction about offshore structures

Ferrant

Touzé

Pelletier

2003

Numerical Methods in Fluids

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SUMMARYThis paper is devoted to the numerical simulation of non-linear wave di raction by three-dimensional (3D) surface-piercing structures in the time domain. Two di erent methods are presented. First a secondorder di raction model is described, in which a boundary-element method combined with a time-stepping procedure are used to solve the ÿrst-and second-order di raction problems in the time domain. The Stokes expansion approach of the free surface non-linearities results in relatively moderate simulation times, so that long simulations in irregular incident waves become feasible. The model has previously been systematically validated by comparison with available semi-analytical frequency domain results on regular wave di raction about simpliÿed geometries. In the present paper, the exibility and stability of the time-domain approach to second-order wave di raction are further demonstrated by the simulation of bichromatic wave di raction over a large number of incident wave periods. The second part of the paper addresses the problem of fully non-linear wave di raction. Again, the solution procedure is based on a boundary element method (BEM) solution of the boundary-value problem in the time domain, but the non-linear boundary conditions are accounted for without any approximation this time. In this fully non-linear di raction model, an explicit description of the incident wave is exploited to solve the problem for the di racted ow only. This approach has a number of practical advantages in terms of accuracy and computational e ciency. In previous publications related to this approach stream-function theory was utilized to model non-linear regular incident waves. In this paper, irregular two-dimensional (2D) incident waves are modelled by means of a recently developed fully non-linear time-domain pseudospectral formulation. An original coupling of this 2D pseudospectral model with the 3D non-linear BEM model is proposed, and its e ectiveness is shown on the case of 2D wave packets interacting with a vertical bottom-mounted cylinder.

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A note on finite depth second-order wave–wave interactions

Cited by 137 publications

References 1 publication

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

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

Estimates of Lagrangian transport by surface gravity wave groups: The effects of finite depth and directionality

Non‐linear time‐domain models for irregular wave diffraction about offshore structures

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