The Lagrangian trajectories of neutrally buoyant particles underneath surface gravity wave groups are dictated by two physical phenomena: the Stokes drift results in a net displacement of particles in the direction of propagation of the group, whereas the Eulerian return flow, as described by the multi-chromatic wave theory of Longuet-Higgins & Stewart (1962 J. Fluid Mech. 13 , 481–504. ( doi:10.1017/S0022112062000877 )), transports such particles in the opposite direction. By pursuing a separation of scales expansion, we develop simple closed-form expressions for the net Lagrangian displacement of particles. By comparing the results from the separation of scales expansion at different orders in bandwidth, we study the effect of frequency dispersion on the local Lagrangian transport, which we show can be ignored for realistic sea states.
AbstractTracer dynamics are computed for a shallow two-layer flow in a circular basin subjected to alternating wind-induced circulation. Lagrangian particle tracking is used to model the dynamics of passive tracers in both the upper and lower layers of a flow with distinct two-layer structure. Results show that particle advection becomes chaotic in parts of the flow in both layers where the effect of external forcing is concentrated primarily in the upper layer, with particles in the lower layer less mixed.Keywords: Chaotic mixingdensity-stratificationKranenburg's basinLagrangian trackingshallow flowstwo-layer lakes and reservoirs AcknowledgementsThe research is partly funded by Malaysian Ministry of Higher Education Fundamental Research Grant Scheme (FRGS) and Research Management Institute (RMI), Universiti Teknologi MARA, Malaysia (Ref: FRGS/1/2012/TK03/UiTM/03/6).
<p>We have performed numerical simulations of steep three-dimensional wave groups, formed by dispersive focusing, using the fully-nonlinear potential flow solver <em>OceanWave3D</em>. We find that third-order resonant interactions result in directional energy transfers to higher-wavenumber components, forming steep wave groups with augmented kinematics and a prolonged lifespan. If the wave group is initially narrow banded, <em>quasi-degenerate interactions</em> resembling the instability band of a regular wave train arise, characterised by unidirectional energy transfers and energy transfers along the resonance angle, &#177;35.26&#176;, of the Phillips &#8216;figure-of-eight&#8217; loop. Spectral broadening due to the quasi-degenerate interactions eventually facilitates <em>non-degenerate interactions</em>, which dominate the spectral evolution of the wave group after focus. The non-degenerate interactions manifest primarily as a high-wavenumber sidelobe, which forms at an angle of &#177;55&#176; to the spectral peak. We consider finite-depth effects in the range of deep to intermediate waters (5.592 &#8805; <em>k<sub>p</sub>d</em> &#8805; 1.363), based on the characteristic wavenumber (<em>k<sub>p</sub></em>) and the domain depth (<em>d</em>), and find that all forms of spectral evolution are suppressed by depth. However, the quasi-degenerate interactions exhibit a greater sensitivity to depth, suggesting suppression of the modulation instability by the return current, consistent with previous studies. We also observe sensitivity to depth for <em>k<sub>p</sub>d</em> values commonly considered "deep", indicating that the length scales of the wave group and return current may be better indicators of dimensionless depth than the length scale of any individual wave component. The non-degenerate interactions appear to be depth resilient with persistent evidence of a &#177;55&#176; spectral sidelobe at a depth of <em>k<sub>p</sub>d</em> =1.363. Although the quasi-degenerate interactions are significantly suppressed by depth, the interactions do not entirely disappear for <em>k<sub>p</sub>d</em> =1.363 and show signs of biasing towards oblique, rather than unidirectional, wave components at intermediate depths. The contraction of the wavenumber spectrum in the <em>k<sub>y</sub></em>-direction has also proved to be resilient to depth, suggesting that lateral expansion of the wave group and the "wall of water" effect of Gibbs & Taylor (2005) may persist at intermediate depths.</p>
Abstract A numerical tool is developed to predict the heading of a weather-vaning FPSO and the freeboard exceedance distribution around an FPSO for the assessment of greenwater overtopping. The heading prediction is compared with full-scale measurements from an operating FPSO for validation. The validated model is then applied to explore the behavior of FPSOs during the passage of tropical cyclones off the North West Shelf (NWS) of Australia. A synthetic database is used to represent tropical cyclone metocean conditions. The vessel is found to experience oblique wave attack frequently due to the misalignment in waves, current and wind. At some instances the vessel can be exposed to beam seas during the passage of a cyclone. Analysis of freeboard exceedance indicates that three locations around the vessel are susceptible to increased risk of greenwater, these are the bow area, amidships and the stern of the vessel. It is also found that for locations around the bow and stern areas of the vessel, combinations of slightly smaller wave heights and shorter wave periods can lead to larger freeboard exceedance events. The freeboard exceedance statistics are sensitive to the vessel heading for the locations on the vessel where the exceedance event occurs. The incoming waves associated with extreme relative wave motion, i.e. freeboard exceedance, in directionally spread seas typical of tropical cyclones are identified. It is found that the maximum elevations of these incoming waves occur some time (on the order of one wave period) before the maximum freeboard exceedance for the seastates and vessel locations analysed due to the vessel motions.
Tertiary interactions between irregular incident waves and their reflections from a rectangular box are investigated experimentally. New experiments feature uni-directional seas with normal and oblique incidence to the side of the box and spread seas, where data is notably lacking in the literature. NewWave conditioning analysis is applied to show that the delays in maximum response associated with tertiary interactions are present in front of the box for both uni-directional and spread sea states. However, only in the uni-directional sea states with normal incidence is there significant amplification of free-surface elevation in front of the box beyond the linear diffraction levels. The nature of the large nonlinear run-up in this case is studied in detail. As an important finding, this work shows that localised tertiary interactions, whilst an interesting physical phenomenon, are unlikely to be important for wave–structure interactions in general, in realistic open ocean wave conditions.
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